Results by chapter On Hilbert’s 6th Problem
The results of On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation that are new in the book, that come from the authors’ own papers, or that correct one of those papers, as the book states them: 92 entries (59 proved, 11 conditional, 21 open, 1 counterexample). Results of other authors that the book presents are credited in the chapter summaries.
Status: proved; conditional (proved assuming the named statements, which the book does not prove); open (stated as an open question); counterexample (a proved counterexample); withdrawn (found wrong after release; the entry says what was wrong and which version corrects it).
Chapter 7. Scale Mixtures, Thorin Measures and Shot Noise
Pólya Gamma integral identity for the logistic likelihood
Theorem 7.5, page 85 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. First stated in Polson, Scott and Windle (2013), Bayesian inference for logistic models using Pólya Gamma latent variables, Journal of the American Statistical Association 108, 1339 to 1349.
Status: proved.
Statement. For \(b>0\) and \(c\in\mathbb{R}\), the Pólya Gamma law \(PG(b,c)\) is the law of \(\omega=\frac{1}{2\pi^2}\sum_{k=1}^\infty \frac{g_k}{(k-\frac12)^2+c^2/(4\pi^2)}\), where \(g_1,g_2,\dots\) are independent Gamma variables with shape \(b\) and unit rate. Let \(p(\cdot\mid b,0)\) be the density of \(PG(b,0)\). Then for all \(b>0\), \(a\in\mathbb{R}\) and \(\psi\in\mathbb{R}\), \(\frac{(e^{\psi})^a}{(1+e^{\psi})^b}=2^{-b}e^{(a-b/2)\psi}\int_0^\infty e^{-\omega\psi^2/2}\,p(\omega\mid b,0)\,\mathrm{d}\omega\). Moreover, the conditional law of \(\omega\) given \(\psi\), that is the law with density proportional to \(e^{-\omega\psi^2/2}p(\omega\mid b,0)\) in \(\omega\), is \(PG(b,\psi)\).
Source: source/chapters/ch05_mixtures.tex, label thm:pg_identity.
Horseshoe density bounds
Theorem 7.9, page 89 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. First stated in Carvalho, Polson and Scott (2010), The horseshoe estimator for sparse signals, Biometrika 97, 465 to 480.
Status: proved.
Statement. Let \(\theta\mid\lambda\sim N(0,\lambda^2)\) with \(\lambda\) half Cauchy \(C^+(0,1)\) (the horseshoe prior with global scale \(\tau=1\)). The marginal density of \(\theta\) is \(p(\theta)=K\,e^{\theta^2/2}E_1(\theta^2/2)\), where \(K=1/\sqrt{2\pi^3}\) and \(E_1(x)=\int_x^\infty s^{-1}e^{-s}\,\mathrm{d}s\) is the exponential integral, and it satisfies, for all \(\theta\neq0\), \(\frac K2\log\big(1+\frac4{\theta^2}\big)<p(\theta)<K\log\big(1+\frac2{\theta^2}\big)\). In particular \(p\) has a logarithmic pole at zero and tails of order \(\theta^{-2}\).
Source: source/chapters/ext/ch05.tex, label thm:horseshoe_bounds.
Chapter 14. The Inverse Problem on the Solution Manifold
Nondegeneracy of the Gaussian limit of the posterior trajectory for reaction diffusion equations
Lemma 14.3, page 189 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Work in the setting of Theorem 1 of Nickl (2024), the Bernstein von Mises theorem for forward trajectories: \(\Omega=[0,1]^d\) is the \(d\) dimensional torus with \(d\le3\), \(T>0\), \(f\in C_c^\infty(\mathbb{R})\), \(\theta_0\) is a smooth initial condition, \(u_\theta\) is the solution of \(\partial_tu-\Delta u=f(u)\), \(u(0)=\theta\), which is observed with standard normal noise at design points uniform on \([0,T]\times\Omega\), and \(\theta\) has the rescaled Gaussian series prior of Theorem 1 of Nickl (2024), of integer regularity \(\gamma>2+3d\). Let \(\mathbb{I}_{\theta_0}\) be the linearization of \(\theta\mapsto u_\theta\) at \(\theta_0\), \(\mathbb{I}_{\theta_0}^*\) its adjoint with respect to Lebesgue measure on \(\Omega\) and on \([0,T]\times\Omega\), and \(\mathcal{N}_{\theta_0}=N(0,T(\mathbb{I}_{\theta_0}^*\mathbb{I}_{\theta_0})^{-1})\) the Gaussian limit of the posterior of \(\theta\). Let \(0<t_{\min}<t_{\max}<\infty\), let \(\mathscr{C}=C([t_{\min},t_{\max}],C(\Omega))\) with the supremum norm, and let \(\mu\) be the law in \(\mathscr{C}\) of the solution \(U\) of \(\partial_tU-\Delta U-f'(u_{\theta_0})U=0\) with \(U(0)=\vartheta\sim\mathcal{N}_{\theta_0}\). Then the reproducing kernel Hilbert space of the Gaussian measure \(\mu\) is a nontrivial linear subspace of \(\mathscr{C}\), and \(r\mapsto\mu(\|U\|_{\mathscr{C}}\le r)\) is continuous and strictly increasing on \([0,\infty)\).
Source: source/chapters/ext/m1a.tex, label m:lem:nondeg.
Manifold gain for the heat equation: prior on the initial condition against a space time Gaussian process prior
Proposition 14.5, page 193 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider the heat equation \(\partial_tu=\nu\Delta u\) with viscosity \(\nu>0\) on the torus \(\mathbb{T}^d=[0,1]^d\), where \(\lambda_j\) are the eigenvalues of \(-\Delta\) (\(\lambda_j=4\pi^2|k_j|^2\) for the Fourier mode \(k_j\in\mathbb{Z}^d\setminus\{0\}\)), \(\lambda_1\) is the smallest, and \((e_j)\) is the corresponding orthonormal eigenbasis. There are \(N\) observations at design points uniform on \([0,T]\times\mathbb{T}^d\) with noise variance \(\varsigma^2\), the variance that the model normalizes to one and that is carried as a symbol only to show where it enters. In the constrained model the prior is placed on the initial condition, with independent Gaussian coefficients of variances \(\sigma_{0,j}^2=\rho^2(\lambda_j/\lambda_1)^{-\alpha}\), where \(\alpha>(d+1)/2\), so that the field is \(\sum_j\theta_je^{-\nu\lambda_jt}e_j\). In the unconstrained model, for comparison, the prior is a Gaussian process on the space time field \(u(t,x)\) with Fourier variances \(\sigma_{0,km}^2=\rho^2((\lambda_k+\omega_m^2)/\lambda_1)^{-\alpha}\), \(\omega_m=2\pi m/T\), \(m\in\mathbb{Z}\), where \(k\) runs over the spatial modes (those with \(\lambda_k\ge\lambda_1\)), so that the two priors have comparable regularity in \(x\). The Gram matrix of the design is replaced by its limit: \(\operatorname{diag}(\iota_j)\) with \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\) in the constrained model, and the identity in the unconstrained one. Then: (i) the constrained posterior standard deviation of \(u(t,x)\) is \(\big(\sum_je^{-2\nu\lambda_jt}(N\iota_j/\varsigma^2+\sigma_{0,j}^{-2})^{-1}\big)^{1/2}\), decreasing in \(t\), and the effective number of parameters, \(\#\{j:N\iota_j\sigma_{0,j}^2\ge\varsigma^2\}\), is, for \(\varsigma=1\), of order \((N\rho^2/\nu T)^{d/(2\alpha+2)}\) as \(\Lambda=(N\rho^2\lambda_1^\alpha/(2\nu T))^{1/(\alpha+1)}\to\infty\) with \(\nu T\Lambda\geq1\), with constants that depend only on \(\alpha\), \(d\) and \(\rho\); (ii) the unconstrained posterior standard deviation is \(\big(\sum_{k,m}(N/\varsigma^2+\sigma_{0,km}^{-2})^{-1}\big)^{1/2}\), independent of \((t,x)\), and its effective number of parameters, \(\#\{(k,m):N\sigma_{0,km}^2\ge\varsigma^2\}\), is of order \(N^{(d+1)/(2\alpha)}\) as \(N\to\infty\); (iii) the ratio of the unconstrained to the constrained standard deviation at time \(t\) is at least \(e^{\nu\lambda_1t}\) times its value at \(t=0\), so it grows without bound as \(t\) increases, at the rate at which the manifold contracts.
Source: source/chapters/ext/m1a.tex, label m:prop:gain.
Chapter 15. Adjoint Sufficient Statistics
Adjoint representation of the sufficient statistic
Lemma 15.1, page 206 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \((e_j)\) be an orthonormal basis of \(L^2(\Omega)\), let \(S_t\) be a strongly continuous semigroup on \(L^2(\Omega)\) with \(S_te_j=s_j(t)e_j\) and generator \(A\), let \((t_i,x_i)\in[0,T]\times\Omega\) and \(Y_i\in\mathbb{R}\) for \(i\le N\), let \(\Phi_{ij}=s_j(t_i)e_j(x_i)\) (\(i\le N\), \(j\le D\)) be the design matrix, so that \((\Phi^\top Y)_j=\sum_{i\le N}Y_is_j(t_i)e_j(x_i)\), and let \(r=\sum_{i\le N}Y_i\,\delta_{t_i}\otimes\delta_{x_i}\) be the empirical data measure on \([0,T]\times\Omega\). Then for every \(h=\sum_jh_je_j\in\operatorname{span}\{e_j:j\le D\}\), \(\langle\Phi^\top Y,h\rangle_{\mathbb{R}^D}=\sum_{i=1}^NY_i\,(S_{t_i}h)(x_i)=\big\langle h,\int_0^TS_t^*r(t)\,\mathrm{d}t\big\rangle\), so that \(\Phi^\top Y\) is the coefficient vector of \(w(0)\), where \(w\) is the mild solution of the backward problem \(-\partial_sw=A^*w+r\), \(w(T)=0\).
Source: source/chapters/ext/m2a.tex, label m:lem:adjoint.
Exact sufficiency, linear case
Proposition 15.2, page 207 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider the linear model \(Y=\Phi\theta+\varepsilon\) with \(\theta\in\mathbb{R}^D\), design matrix \(\Phi_{ij}=s_j(t_i)e_j(x_i)\) (\(i\le N\), \(j\le D\), where \(S_te_j=s_j(t)e_j\) for a semigroup that is diagonal in the basis \((e_j)\)), noise \(\varepsilon\sim N(0,\mathrm{Id}_N)\) and Gaussian prior \(\theta\sim N(0,\Sigma_0)\). Let \(S_N=\Phi^\top Y\) be the adjoint statistic (the coefficient vector of \(w(0)\), where \(w\) solves the backward adjoint problem driven by the data) and \(V_N=(\Phi^\top\Phi+\Sigma_0^{-1})^{-1}\). Then \(S_N\) together with the design is sufficient for \(\theta\); the posterior given \(S_N\) is exactly the posterior given all the data, namely \(N(V_NS_N,V_N)\); and the conditional quantile map is exactly affine: for each coordinate \(j\) and \(\tau\in(0,1)\), \(Q_j(\tau\mid S_N)=(V_NS_N)_j+z_\tau(V_N)_{jj}^{1/2}\), where \(z_\tau\) is the \(\tau\) quantile of the standard normal law. The slope of this map in \(S_N\) is \(V_N\), not the identity. For the preconditioned summary \(\tilde S_N=V_NS_N\), which is the posterior mean, the map \(\tilde S_N\mapsto Q_j(\tau\mid S_N)\) has slope exactly one and, apart from the data independent scale term \(z_\tau(V_N)_{jj}^{1/2}\), intercept zero. Consequently generative Bayesian computation (GBC) with summary \(\tilde S_N\) and an identity plus scale generator is exact for every \(N\), up to the estimation of the \(D\times D\) scale from \(M\) simulations; in the linear model that scale is \(V_N\) itself and is known in closed form.
Source: source/chapters/ext/m2a.tex, label m:prop:linsuff.
Equivalence of the one step efficient score summary with the efficient centering
Proposition 15.5, page 210 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\Omega=[0,1]^d\) be the torus, \(T>0\), \(\mathcal X=[0,T]\times\Omega\), and let \(\mathscr{G}(\theta)=u_\theta\) be the solution map of an evolution equation with initial condition \(\theta\) and values in \(W\). The data are \(Z^{(N)}=(Y_i,X_i)_{i\le N}\) with \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\), \(X_i\) i.i.d. uniform on \(\mathcal X\) and \(\varepsilon_i\sim N(0,I_W)\) independent. Let \((e_j)\) be the orthonormal Laplace eigenbasis of mean zero functions, with eigenvalues \(\lambda_j\asymp j^{2/d}\); let the prior \(\Pi\) be the law of \((\sqrt N\delta_N)^{-1}\sum_j\lambda_j^{-\gamma/2}g_je_j\) with \(g_j\) i.i.d. \(N(0,1)\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\), and \(\Pi_N=\Pi(\cdot\mid Z^{(N)})\) the posterior; let \(\theta_0\) be the true initial condition, \(P^N_{\theta_0}\) the law of the data, \(\mathbb{I}_\vartheta\) the linearization of \(\mathscr{G}\) at \(\vartheta\), \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1})\), \(J_N\asymp N\delta_N^2\) and \(p_N\) the projection onto \(\operatorname{span}\{e_j:j\le J_N\}\).
Suppose Conditions 1 and 2 of Nickl (2024) hold with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\) as in his Bernstein von Mises theorem (\(\eta_0\) the index of his invertibility condition), that is, the prior above with Sobolev reproducing kernel space \(H^\gamma\) and, for the forward map on balls of \(H^{\bar\gamma}\), uniform \(L^\infty\) bounds, \(L^2\) Lipschitz continuity, a linearization at \(\theta_0\) with quadratic remainder, \(L^\infty\) mapping properties from \(H^\zeta\) with \(\zeta>d/2\), the stability estimate \(\|\mathscr{G}(\theta)-\mathscr{G}(\theta_0)\|_{L^2(\mathcal X)}\ge c\|\theta-\theta_0\|_{H^{-1}}\) and an invertible information operator (\(\bar\gamma\) is the Sobolev index of the localization ball, \(\max(2,2\zeta)<\bar\gamma<\gamma-\max(1,d/2)\)). Let \(\kappa\) satisfy \(2d+\zeta<\kappa<k-d/2\).
The one step efficient score summary is defined by cross fitting. The sample is split into two halves with index sets \(\mathcal I_1,\mathcal I_2\) of sizes differing by at most one. For \(b=1,2\) a preliminary estimate \(\bar\theta^{[b]}_N\) is computed from half \(b\) alone, takes values in the parameter space and satisfies \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for \(0\le\xi\le\bar\gamma\), where \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\). With \(\bar e_j^{\,\vartheta}=T(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1}e_j\), put for \(j\le J_N\) \(S_{N,j}=\frac12\sum_{b=1,2}\big[\langle e_j,\bar\theta^{[b]}_N\rangle+\frac1{|\mathcal I_{3-b}|}\sum_{i\in\mathcal I_{3-b}}\langle\mathbb{I}_{\bar\theta^{[b]}_N}p_N(\bar e_j^{\,\bar\theta^{[b]}_N})(X_i),\,Y_i-u_{\bar\theta^{[b]}_N}(X_i)\rangle_W\big]\) and \(\hat S^{J_N}_N=\sum_{j\le J_N}S_{N,j}e_j\). The efficient centering of Nickl’s proof is \(\hat\theta_N=\sum_{j\ge1}\hat\Psi_{N,j}e_j\) with \(\hat\Psi_{N,j}=\langle e_j,\theta_0\rangle+\frac1N\sum_{i=1}^N\langle\mathbb{I}_{\theta_0}p_N(\bar e_j^{\,\theta_0})(X_i),\varepsilon_i\rangle_W\).
Suppose further that \(\sqrt N\,\tilde\delta_N(0)\tilde\delta_N(\zeta)\to0\); that \(\gamma<(\bar\gamma+1)(\kappa-d/2)\) unless \(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) is diagonal in \((e_j)\) or the preliminary estimates converge at the rate \(\delta_N\) in \(H^{-1}\); and that the weights depend on the linearization point in a Lipschitz way: for every \(B>0\) there is \(L=L(B)\) such that for all \(\vartheta\) with \(\|\vartheta\|_{H^{\bar\gamma}}\le B\) and all \(j\le J_N\), \(\|\mathbb{I}_\vartheta p_N(\bar e_j^{\,\vartheta})-\mathbb{I}_{\theta_0}p_N(\bar e_j^{\,\theta_0})\|_{L^2(\mathcal X)}\le L\,\|\vartheta-\theta_0\|_{H^\zeta}\|e_j\|_{H^{\kappa+2}}\). Then \(\sqrt N\,\|\hat S^{J_N}_N-\hat\theta_N\|_{H^{-k-2}}\to0\) in \(P^N_{\theta_0}\) probability, and consequently \(\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-\hat S^{J_N}_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\to0\) in \(P^N_{\theta_0}\) probability, where \(\mathscr{W}_1\) is the Wasserstein distance of order one on \(H^{-k-2}\).
Source: source/chapters/ext/m2a.tex, label m:prop:suff.
The one step summary in Sobolev norms
Proposition 15.6, page 213 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In Nickl’s framework, for the cross fitted one step efficient score summary of the book (torus \(\Omega=[0,1]^d\), data \(Y_i=\mathscr{G}(\theta_0)(X_i)+\varepsilon_i\) at the true initial condition \(\theta_0\) with \(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\) and \(\varepsilon_i\sim N(0,I_W)\), orthonormal Laplace eigenbasis \((e_j)\) with eigenvalues \(\lambda_j\asymp j^{2/d}\), rescaled Gaussian series prior with exponent \(\gamma\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(J_N\asymp N\delta_N^2\), \(p_N\) the projection onto the first \(J_N\) modes, \(\mathbb{I}_\vartheta\) the linearization of \(\mathscr{G}\) at \(\vartheta\), \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\)), suppose that Conditions 1 and 2 of Nickl (2024) hold (the prior and, for the forward map on balls of \(H^{\bar\gamma}\), uniform bounds, a linearization at \(\theta_0\) with quadratic remainder, the stability estimate \(\|\mathscr{G}(\theta)-\mathscr{G}(\theta_0)\|_{L^2(\mathcal X)}\ge c\|\theta-\theta_0\|_{H^{-1}}\) and an invertible information operator). Suppose that for every \(B>0\) there are constants \(0<c_B\le C_B<\infty\) such that for every \(\vartheta\) in the parameter space with \(\|\vartheta\|_{H^{\bar\gamma}}\le B\), \(c_B\|h\|_{H^{-1}}\le\|\mathbb{I}_\vartheta h\|_{L^2(\mathcal X)}\le C_B\|h\|_{H^{-1}}\) for all \(h\in L^2_0(\Omega)\) (with \(\dot H^{-1}_\diamond\) and \(\dot H_\diamond\) in place of \(H^{-1}\) and \(L^2_0(\Omega)\) for the Stokes and Navier Stokes equations), and that the preliminary estimates \(\bar\theta^{[b]}_N\) of the cross fitted one step summary (each computed from one half sample alone, with values in the parameter space and \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for \(0\le\xi\le\bar\gamma\)) satisfy \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^{-1}}=O_P(\delta_N)\) for \(b=1,2\). Then the projected one step statistic \(\hat S^{J_N}_N\) satisfies \(\|\hat S^{J_N}_N-\theta_0\|_{H^s}=O_P\big(N^{(s+1-\gamma)/(2\gamma+d)}\big)\) for every \(-1\le s\le\gamma\). If moreover \(\bar\gamma<\gamma-1\), then \(\|\hat S^{J_N}_N-\theta_0\|_{L^2}=o_P(\tilde\delta_N(0))\) and \(\|\hat S^{J_N}_N-\theta_0\|_{H^{\bar\gamma}}=o_P(1)\). For a linear forward map whose information operator is diagonal in \((e_j)\), as for the heat and Stokes equations, the same conclusions hold without the rate in \(H^{-1}\).
Source: source/chapters/ext/m2a.tex, label m:prop:sumrate.
Preliminary estimates with the rate in \(H^{-1}\)
Lemma 15.7, page 215 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In Nickl’s framework (data \(Y_i=\mathscr{G}(\theta_0)(X_i)+\varepsilon_i\) at the true initial condition \(\theta_0\) on \(\mathcal X=[0,T]\times\Omega\) with uniform design and standard Gaussian noise, rescaled Gaussian series prior with exponent \(\gamma\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\)), suppose that Conditions 1 and 2 of Nickl (2024) hold with \(\gamma\) and \(k\) as in his Bernstein von Mises theorem, and that \(\bar\gamma<\gamma-d/2\). Let \(\bar\theta^{[b]}_N\) be computed from the half sample \(Z^{[b]}\) alone, with values in the parameter space and \(\|\bar\theta^{[b]}_N\|_{H^{\bar\gamma}}=O_P(1)\). If \(\|u_{\bar\theta^{[b]}_N}-u_{\theta_0}\|_{L^2(\mathcal X)}=O_P(\delta_N)\), then \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^{-1}}=O_P(\delta_N)\), and this rate in \(H^{-1}\) implies \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for every \(-1\le\xi\le\bar\gamma\). The posterior mean computed from \(Z^{[b]}\) with the prior at the sample size of that half satisfies the rate in \(H^{-1}\) and these rates; for the two dimensional Navier Stokes equations this holds in the setting of Theorem 2.11 of Konen and Nickl (2026) when \(\bar\gamma\le11\).
Source: source/chapters/ext/m2a.tex, label m:lem:halfmean.
Sobolev rates of the one step summary under the efficient centering hypotheses
Corollary 15.9, page 217 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Assume all hypotheses of the equivalence of the one step efficient score summary \(\hat S^{J_N}_N\) with the efficient centering \(\hat\theta_N\) (catalog entry equivalence_with_the_efficient_centering): Conditions 1 and 2 of Nickl (2024) with \(\gamma\) and \(k\) as in his Bernstein von Mises theorem; \(2d+\zeta<\kappa<k-d/2\); \(\sqrt N\,\tilde\delta_N(0)\tilde\delta_N(\zeta)\to0\) with \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\); \(\gamma<(\bar\gamma+1)(\kappa-d/2)\) unless \(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) is diagonal in the Laplace eigenbasis \((e_j)\) or the preliminary estimates converge at the rate \(\delta_N\) in \(H^{-1}\); and the Lipschitz dependence of the weights on the linearization point, \(\|\mathbb{I}_\vartheta p_N(\bar e_j^{\,\vartheta})-\mathbb{I}_{\theta_0}p_N(\bar e_j^{\,\theta_0})\|_{L^2(\mathcal X)}\le L(B)\|\vartheta-\theta_0\|_{H^\zeta}\|e_j\|_{H^{\kappa+2}}\) for all \(\|\vartheta\|_{H^{\bar\gamma}}\le B\) and \(j\le J_N\), where \(\bar e_j^{\,\vartheta}=T(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1}e_j\). Then for every \(-1\le s\le\gamma\), \(\|\hat S^{J_N}_N-\theta_0\|_{H^s}=O_P\big(N^{(s+1-\gamma)/(2\gamma+d)}\big)+o_P\big(N^{(s+k+2)/(2\gamma+d)-1/2}\big)\).
Source: source/chapters/ext/m2a.tex, label m:cor:sumsuff.
The information operator of the reaction diffusion class
Lemma 15.11, page 219 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider the reaction diffusion equation \(\partial_tu-\Delta u=f(u)\), \(u(0)=\theta\), on the torus \(\Omega=[0,1]^d\) with \(f\in C_c^\infty(\mathbb{R})\) and \(d\le3\) (the class of Theorem 1 of Nickl (2024)), with smooth true initial condition \(\theta_0\), \(\zeta>d/2\) an index for which his Condition 2D holds, solution map \(\mathscr{G}\), linearization \(\mathbb{I}_\vartheta\) at \(\vartheta\), parameter space the Sobolev space \(H^{\bar\gamma}\) with \(\bar\gamma>3+d/2\) (Theorem 5 of Nickl (2024) verifies his Condition 2 for this class), and \(\mathcal X=[0,T]\times\Omega\). Let \(\mathcal{L}\) be the solution operator of the heat equation \(\partial_tu=\Delta u\) and \(\mathcal{K}_\vartheta=\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta-\mathcal{L}^*\mathcal{L}\), both compressed to \(L^2_0(\Omega)\). For every \(B>0\) there is a constant \(C\), depending on \(B\), \(f\), \(T\) and \(d\), such that for all \(\vartheta\) and \(\vartheta+h\) with \(H^{\bar\gamma}\) norms at most \(B\): (i) \(|\langle\mathcal{K}_\vartheta\phi,\psi\rangle|\le C\|\phi\|_{H^{-1}}\|\psi\|_{H^{-3}}\) for \(\phi,\psi\in L^2_0(\Omega)\), so that the symmetric operator \(\mathcal{K}_\vartheta\) extends to maps from \(H^{-1}_0\) to \(H^3_0\) and from \(H^{-3}_0\) to \(H^1_0\) with norms at most \(C\); (ii) \(\|\mathscr{G}(\vartheta+h)-\mathscr{G}(\vartheta)-\mathbb{I}_\vartheta h\|_{L^2(\mathcal X)}\le C\|h\|_{L^2}^2\); (iii) the weights of the one step summary satisfy the Lipschitz condition \(\|\mathbb{I}_\vartheta p_N(\bar e_j^{\,\vartheta})-\mathbb{I}_{\theta_0}p_N(\bar e_j^{\,\theta_0})\|_{L^2(\mathcal X)}\le L\,\|\vartheta-\theta_0\|_{H^\zeta}\|e_j\|_{H^{\kappa+2}}\) for every \(\kappa\ge-1\), with \(\bar e_j^{\,\vartheta}=T(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1}e_j\), \(p_N\) the projection onto the first \(J_N\) Laplace eigenfunctions \(e_j\), and \(L\) depending on \(B\).
Source: source/chapters/ext/m2a.tex, label m:lem:rdinfo.
The one step summary in Sobolev norms: reaction diffusion
Proposition 15.12, page 219 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the setting of the information operator lemma for the reaction diffusion class (the equation \(\partial_tu-\Delta u=f(u)\) with \(f\in C_c^\infty(\mathbb{R})\), \(d\le3\), parameter space \(H^{\bar\gamma}\) with \(\bar\gamma>3+d/2\), Conditions 1 and 2 of Nickl (2024) as verified in his Theorem 5, integer exponent \(\gamma>2+3d\) of the rescaled Gaussian series prior, smooth true initial condition \(\theta_0\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(J_N\asymp N\delta_N^2\)) let \(\bar\gamma<\gamma-1\). Then for every choice of the preliminary estimates that the definition of the cross fitted one step summary allows (computed on each half sample, with values in the parameter space and \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for \(0\le\xi\le\bar\gamma\), \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\)), the projected one step statistic satisfies \(\|\hat S^{J_N}_N-\theta_0\|_{L^2}=o_P(\tilde\delta_N(0))\) and \(\|\hat S^{J_N}_N-\theta_0\|_{H^{\bar\gamma}}=o_P(1)\).
Source: source/chapters/ext/m2a.tex, label m:prop:sumrd.
The regularized summary on \(J_N\) modes: heat and Stokes
Proposition 15.13, page 220 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\mathscr{G}\) be the solution map of the heat equation \(\partial_tu=\nu\Delta u\) on \(\mathbb{T}^d\), with \(W=\mathbb{R}\), or of the Stokes equations on \(\mathbb{T}^2\), with \(W=\mathbb{R}^2\) and \(d=2\), and let \((e_j)\) be a real orthonormal eigenbasis of the mean zero functions, respectively of the divergence free mean zero fields, with \(\sup_x|e_j(x)|_W\le\sqrt2\), as for the trigonometric basis; thus \(s_j(t)=e^{-\nu\lambda_jt}\) and \(\lambda_j\asymp j^{2/d}\). The data are \(Z^{(N)}=(Y_i,X_i)_{i\le N}\) with \(Y_i=\mathscr{G}(\theta_0)(X_i)+\varepsilon_i\), \(X_i=(t_i,x_i)\) i.i.d. uniform on \([0,T]\times\mathbb{T}^d\) and \(\varepsilon_i\sim N(0,I_W)\), where \(\theta_0\in H^\gamma\) lies in the closed span of \((e_j)\) and \(\gamma>d+1\) is the exponent of the prior \(\Pi\), the law of \((\sqrt N\delta_N)^{-1}\sum_j\lambda_j^{-\gamma/2}g_je_j\) with \(g_j\) i.i.d. \(N(0,1)\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\); let \(k>d/2\), \(J_N\asymp N\delta_N^2\) and \(\iota_j=\frac1T\int_0^Ts_j(t)^2\,\mathrm{d}t\). Retain \(D=D_N\) modes and give their coefficients the truncation of the prior, \(N(0,\Sigma_0)\) with \(\Sigma_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le D}\) and \(\sigma_{0,j}^2=\lambda_j^{-\gamma}/(N\delta_N^2)\); write \(\Pi^D(\cdot\mid Z^{(N)})\) for the resulting posterior and \(\Phi\) for the \(N\times D\) design matrix with entries \(s_j(t_i)e_j(x_i)\) (a pair of rows per observation for Stokes). Suppose that \(D_N\to\infty\), \(\delta_N^2D_N^{2+2/d}\to0\) and \(ND_N^{-2(k+2+\gamma)/d}\to0\), which holds for \(D_N\asymp J_N\). Let \(\hat S^{\rm reg}_N\) be the regularized one step summary computed on the \(D\) retained modes from a pilot \(\bar\theta_N\) in their span, which may depend on the data in any way: \(\hat S^{\rm reg}_N=\bar\theta_N+(F_N+\Sigma_0^{-1})^{-1}\big(\Phi^\top(Y-\Phi\bar\theta_N)-\Sigma_0^{-1}\bar\theta_N\big)\) with \(F_N=\Phi^\top\Phi\), which equals \((\Phi^\top\Phi+\Sigma_0^{-1})^{-1}\Phi^\top Y\), the posterior mean under the truncated prior, for every pilot. Let \(\hat\theta_N=\theta_0+\sum_{j\le J_N}\xi_je_j\) with \(\xi_j=(N\iota_j)^{-1}\sum_{i\le N}s_j(t_i)\langle e_j(x_i),\varepsilon_i\rangle_W\) be the efficient centering of Nickl (2024) for these equations, \(\mathcal{N}_{\theta_0}\) the Gaussian law with covariance \(\operatorname{diag}(1/\iota_j)\) in the basis \((e_j)\), and \(\mathscr{W}_1\) the Wasserstein distance of order one on \(H^{-k-2}\). Then, in \(P^N_{\theta_0}\) probability where convergence in probability is meant:
- \(\sqrt N\|\hat S^{\rm reg}_N-\hat\theta_N\|_{H^{-k-2}}\to0\) in probability; (ii) \(\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-\hat S^{\rm reg}_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\to0\) in probability for \(\theta\sim\Pi^D(\cdot\mid Z^{(N)})\); (iii) \(\sqrt N(\hat S^{\rm reg}_N-\theta_0)\to\mathcal{N}_{\theta_0}\) in distribution in \(H^{-k-2}\); (iv) for the heat equation with integer \(\gamma>3d+2\) and \(k>3d\), Nickl’s Bernstein von Mises theorem applies, (ii) holds also for \(\theta\) drawn from the posterior \(\Pi_N\) under the untruncated prior, and the mean \(\tilde\theta_N\) of \(\Pi_N\) satisfies \(\sqrt N\|\hat S^{\rm reg}_N-\tilde\theta_N\|_{H^{-k-2}}\to0\) in probability; (v) for the heat equation, whenever the hypotheses of the equivalence of the cross fitted one step summary \(\hat S^{J_N}_N\) with the efficient centering hold (catalog entry equivalence_with_the_efficient_centering), \(\sqrt N\|\hat S^{\rm reg}_N-\hat S^{J_N}_N\|_{H^{-k-2}}\to0\) in probability.
Source: source/chapters/ext/m2a.tex, label m:prop:regheat.
The regularized summary under a fixed prior, and the untruncated posteriors
Proposition 15.14, page 225 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let the setting be that of the regularized summary on \(D=D_N\) modes for the heat equation \(\partial_tu=\nu\Delta u\) on \(\mathbb{T}^d\) (\(W=\mathbb{R}\)) or the Stokes equations on \(\mathbb{T}^2\) (\(W=\mathbb{R}^2\), \(d=2\)): \((e_j)\) a real orthonormal eigenbasis of the mean zero functions, respectively divergence free mean zero fields, with \(\sup_x|e_j(x)|_W\le\sqrt2\); \(s_j(t)=e^{-\nu\lambda_jt}\); data \(Y_i=\mathscr{G}(\theta_0)(X_i)+\varepsilon_i\) with \(X_i\) i.i.d. uniform on \([0,T]\times\mathbb{T}^d\) and \(\varepsilon_i\sim N(0,I_W)\); \(\iota_j=\frac1T\int_0^Ts_j(t)^2\,\mathrm{d}t\); \(\hat S^{\rm reg}_N=(\Phi^\top\Phi+\Sigma_0^{-1})^{-1}\Phi^\top Y\) the regularized summary (posterior mean under the truncated prior) with \(\Phi_{ij}=s_j(t_i)e_j(x_i)\); \(\hat\theta_N=\theta_0+\sum_{j\le J_N}\xi_je_j\) with \(\xi_j=(N\iota_j)^{-1}\sum_is_j(t_i)\langle e_j(x_i),\varepsilon_i\rangle_W\) the efficient centering; \(\mathscr{W}_1\) the Wasserstein distance of order one on \(H^{-k-2}\); \(\mathcal{N}_{\theta_0}\) the Gaussian law with covariance \(\operatorname{diag}(1/\iota_j)\). Take now \(\theta_0\in H^\beta\) in the closed span of \((e_j)\) with \(\beta>d/2\), \(k>d/2-1\), and give the \(D=D_N\) retained coefficients the fixed prior \(\sigma_{0,j}^2=\rho^2\lambda_j^{-\alpha}\) with \(\rho>0\) and \(\alpha>d/2\) (not rescaled with \(N\)). Suppose that \(\beta\ge\alpha-k-1\), that \(D_N\to\infty\) with \(\sup_NND_N^{-2(k+2+\beta)/d}<\infty\) (a condition void when \(\theta_0\) has finitely many nonzero coefficients), and that \(\alpha>d\) or \(D_N^{2+2/d}/N\to0\), and form the centering \(\hat\theta_N\) with the first \(J_N\) modes, for any \(J_N\to\infty\). Then (i) \(\sqrt N\|\hat S^{\rm reg}_N-\hat\theta_N\|_{H^{-k-2}}\to0\) in probability, (ii) \(\mathscr{W}_1(\operatorname{Law}(\sqrt N(\theta-\hat S^{\rm reg}_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0})\to0\) in probability for \(\theta\) drawn from the posterior under the truncated prior, (iii) \(\sqrt N(\hat S^{\rm reg}_N-\theta_0)\to\mathcal{N}_{\theta_0}\) in distribution in \(H^{-k-2}\), and (v) for the heat equation, \(\sqrt N\|\hat S^{\rm reg}_N-\hat S^{J_N}_N\|_{H^{-k-2}}\to0\) in probability whenever the hypotheses of the equivalence of the cross fitted one step summary \(\hat S^{J_N}_N\) with the efficient centering hold. If \(\alpha>d\), (i) to (iii) hold also for \(D=\infty\), that is, under the untruncated prior.
For the rescaled prior \(\sigma_{0,j}^2=\lambda_j^{-\gamma}/(N\delta_N^2)\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(\gamma>d+1\), of the regularized summary on \(J_N\) modes (catalog entry the_regularized_summary_on_modes_heat_and_stokes), the conclusions of that statement hold without its condition \(\delta_N^2D_N^{2+2/d}\to0\), and its claims (i) to (iii) hold also for \(D=\infty\), that is, for the posterior \(\Pi_N\) under the untruncated rescaled prior and its mean \(\tilde\theta_N\). Hence the last two claims of its part (iv) (that (ii) holds for \(\theta\) drawn from \(\Pi_N\) and that \(\sqrt N\|\hat S^{\rm reg}_N-\tilde\theta_N\|_{H^{-k-2}}\to0\) in probability) hold for the Stokes equations as well, and for the heat equation without its restrictions on \(\gamma\) and \(k\); and when \(\theta_0\) meets both sets of hypotheses, the regularized summaries under the two priors, and if \(\alpha>d\) the untruncated posterior means, are \(\sqrt N\) equivalent in \(H^{-k-2}\).
Source: source/chapters/ext/m2a.tex, label m:prop:regfixed.
The regularized summary on \(J_N\) modes: reaction diffusion
Proposition 15.16, page 228 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider the reaction diffusion equation \(\partial_tu-\Delta u=f(u)\), \(u(0)=\theta\), on the torus \(\Omega=[0,1]^d\) with \(f\in C_c^\infty(\mathbb{R})\) and \(d\le3\) (the class of Theorem 1 of Nickl (2024)), with the rescaled Gaussian series prior \(\Pi\) of integer exponent \(\gamma>2+3d\) (\(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(J_N\asymp N\delta_N^2\)), smooth true initial condition \(\theta_0\), parameter space \(H^{\bar\gamma}\) with \(\bar\gamma>3+d/2\), and let \(k>3d\), so that Theorem 5 of Nickl (2024) gives Conditions 1 and 2 with \(\gamma\) and \(k\) as in his Bernstein von Mises theorem. Retain \(D=J_N\) modes with the truncated prior \(\Sigma_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J_N}\), \(\sigma_{0,j}^2=\lambda_j^{-\gamma}/(N\delta_N^2)\). The sample is split into halves with index sets \(\mathcal I_1,\mathcal I_2\); let the preliminary estimates \(\bar\theta^{[b]}_N\) (computed from half \(b\), with the rates of the cross fitted one step summary) lie in the span of \(e_1,\dots,e_{J_N}\), as their projections do with the same rates. For \(b=1,2\) let \(\hat S^{{\rm reg},b}_N=\bar\theta^{[b]}_N+(F^b_N+\Sigma_0^{-1})^{-1}\big(\mathbb{I}^*_{\bar\theta^{[b]}_N}[\sum_{i\in\mathcal I_{3-b}}r_i\,\delta_{X_i}]-\Sigma_0^{-1}\bar\theta^{[b]}_N\big)\) with residuals \(r_i=Y_i-u_{\bar\theta^{[b]}_N}(X_i)\) and \(F^b_N\) the realized information \(\mathbb{I}^*_{\bar\theta^{[b]}_N}\mathbb{I}_{\bar\theta^{[b]}_N}\) on the design points \(X_i\), \(i\in\mathcal I_{3-b}\), compressed to the retained modes, and put \(\hat S^{\rm reg}_N=\frac12(\hat S^{{\rm reg},1}_N+\hat S^{{\rm reg},2}_N)\). Let \(\hat\theta_N\) be the efficient centering of Nickl’s proof, \(\mathcal{N}_{\theta_0}=N(0,T(\mathbb{I}_{\theta_0}^*\mathbb{I}_{\theta_0})^{-1})\), \(\Pi_N\) the posterior, \(\tilde\theta_N\) its mean and \(\mathscr{W}_1\) the Wasserstein distance of order one on \(H^{-k-2}\). Then, in \(P^N_{\theta_0}\) probability: (i) \(\sqrt N\|\hat S^{\rm reg}_N-\hat\theta_N\|_{H^{-k-2}}\to0\); (ii) \(\mathscr{W}_1(\operatorname{Law}(\sqrt N(\theta-\hat S^{\rm reg}_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0})\to0\) for \(\theta\sim\Pi_N\), the mean \(\tilde\theta_N\) satisfies \(\sqrt N\|\hat S^{\rm reg}_N-\tilde\theta_N\|_{H^{-k-2}}\to0\), and \(\sqrt N(\hat S^{\rm reg}_N-\theta_0)\to\mathcal{N}_{\theta_0}\) in distribution in \(H^{-k-2}\); (iii) under the hypotheses of the equivalence of the cross fitted one step summary \(\hat S^{J_N}_N\) with the efficient centering other than the Lipschitz condition on the weights (which the information operator lemma for this class supplies), \(\sqrt N\|\hat S^{\rm reg}_N-\hat S^{J_N}_N\|_{H^{-k-2}}\to0\).
Source: source/chapters/ext/m2a.tex, label m:prop:regrd.
Expected utility form of the Bernstein von Mises limit
Proposition 15.18, page 231 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For the heat or the Stokes equation on the torus, with semigroup eigenvalues \(s_j(t)=e^{-\nu\lambda_jt}\) in the Laplace eigenbasis \((e_j)\) (\(\lambda_j\asymp j^{2/d}\)), data \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) with \(X_i\) i.i.d. uniform on \([0,T]\times\Omega\) and \(\varepsilon_i\sim N(0,I_W)\), and per mode information \(\iota_j=\frac1T\int_0^Ts_j(t)^2\,\mathrm{d}t\), consider the first \(J\) modes with prior \(\Pi^J=N(0,\Sigma_0)\), \(\Sigma_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J}\), in the Gaussian experiment that observes \(\theta_j+(N\iota_j)^{-1/2}\xi_j\), \(j\le J\), with independent standard normal \(\xi_j\). Its information matrix is \(\operatorname{diag}(N\iota_j)\), the population value of the Gram matrix \(\Phi^\top\Phi\) of the design, and it is the \(J\) mode experiment of the Bernstein von Mises limit, whose covariance is \(T(\mathbb{I}^*\mathbb{I})^{-1}/N=\operatorname{diag}(1/(N\iota_j))\), where \(\mathbb{I}\) is the forward map of the equation, which is linear. Write \(\kappa_j=N\iota_j\sigma_{0,j}^2\) for the per mode ratio of information to prior precision, and let \(I_N=\mathbb{E}^\Pi\big[\log\frac{\mathrm{d}\Pi^J(\cdot\mid Z^{(N)})}{\mathrm{d}\Pi^J}\,\big|\,Z^{(N)}\big]\) be the realized information gain, with \(\Pi^J(\cdot\mid Z^{(N)})\) the posterior and \(\tilde\theta^J_N\) the posterior mean. Then for every \(N\) and \(J\), \(I_N=\frac12\sum_{j\le J}\big[\log(1+\kappa_j)-\frac{\kappa_j}{1+\kappa_j}\big]+\frac12\tilde\theta_N^{J\top}\Sigma_0^{-1}\tilde\theta^J_N\), and averaging over the prior predictive law gives the Lindley expected information gain \(\mathbf{E}\,I_N=\frac12\log\det\big(\mathrm{Id}+\frac NT\mathbb{I}^*\mathbb{I}\,\Sigma_0\big)=\frac12\sum_{j\le J}\log(1+\kappa_j)\). In the linear model with \(D=J\) modes, that is, with the realized information \(\Phi^\top\Phi\) in place of \(\operatorname{diag}(N\iota_j)\), both identities hold conditionally on the design, with the \(\kappa_j\) replaced by the eigenvalues of \(\Sigma_0^{1/2}\Phi^\top\Phi\,\Sigma_0^{1/2}\). Every mode contributes the strictly positive amount \(\frac12\log(1+\kappa_j)\), and every mode has posterior variance \(\sigma_{0,j}^2/(1+\kappa_j)\) strictly below its prior variance; the modes with \(\kappa_j>1\) are those whose posterior variance falls below half the prior variance. Under the rescaled prior \(\sigma_{0,j}^2=\lambda_j^{-\gamma}/(N\delta_N^2)\) with \(\delta_N=N^{-\gamma/(2\gamma+d)}\) and \(J=J_N\asymp N\delta_N^2\), the number of these is \(J^{\rm eff}_N=\#\{j\le J_N:\kappa_j>1\}\asymp N^{\gamma d/((\gamma+1)(2\gamma+d))}=J_N^{\gamma/(\gamma+1)}=o(J_N)\), so the experiment is informative in every retained mode, but strongly informative in only a vanishing fraction of them.
Source: source/chapters/ext/m2a.tex, label m:prop:pr.
Exact identities for the information gain under any forward map
Lemma 15.21, page 233 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\mathscr{G}\) be a measurable forward map in the model \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) (design \(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\), noise \(\varepsilon_i\sim N(0,I_W)\)), and let the prior be a centered Gaussian measure under which the coordinates \(\langle\theta,e_j\rangle\) in an orthonormal basis \((e_j)\) are independent with variances \(\sigma_{0,j}^2>0\). For a number \(J\) of modes let \(\theta^J=(\langle\theta,e_j\rangle)_{j\le J}\), let \(\Pi^J\) and \(\Pi^J(\cdot\mid Z^{(N)})\) be its prior and posterior laws, \(\Sigma^J_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J}\), \(m_N\) and \(V_N\) the posterior mean and covariance of \(\theta^J\), \(I_N=\mathrm{KL}\big(\Pi^J(\cdot\mid Z^{(N)})\,\|\,\Pi^J\big)\) the realized information gain, \(\mathbf{P}\) the prior predictive law (parameter \(\vartheta\sim\Pi\), then data \(Z^{(N)}\sim P^N_\vartheta\)) and \(\mathbf{E}\) its expectation. Then for every \(N\) and \(J\): (i) almost surely \(I_N=\mathrm{KL}\big(N(m_N,V_N)\,\|\,\Pi^J\big)+\Gamma_N\), where \(\Gamma_N=\mathrm{KL}\big(\Pi^J(\cdot\mid Z^{(N)})\,\|\,N(m_N,V_N)\big)\ge0\), and \(\Gamma_N\le\mathrm{KL}\big(\Pi^J(\cdot\mid Z^{(N)})\,\|\,G\big)\) for every nondegenerate Gaussian law \(G\) on \(\mathbb{R}^J\); (ii) if \(\int\|\mathscr{G}(\vartheta)\|^2_{L^\infty(\mathcal X)}\,\mathrm{d}\Pi(\vartheta)<\infty\), then \(\mathbf{E}\,I_N=I(\theta^J;Z^{(N)})<\infty\) (the mutual information) and \(\mathbf{E}\,I_N=\frac12\,\mathbf{E}\log\det\big(\Sigma^J_0V_N^{-1}\big)+\mathbf{E}\,\Gamma_N\). In the linear Gaussian models of the heat and Stokes equations the posterior is \(N(m_N,V)\) with \(V\) not depending on the data, so \(\Gamma_N=0\) and (i) and (ii) reduce to the two displays of the expected utility form of the Bernstein von Mises limit.
Source: source/chapters/ext/m2a.tex, label m:lem:prexact.
Kullback Leibler form of the Bernstein von Mises theorem
Theorem 15.22, page 234 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In Nickl’s framework (torus \(\Omega=[0,1]^d\), data \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) with \(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\) and \(\varepsilon_i\sim N(0,I_W)\), rescaled Gaussian series prior \(\Pi\) with exponent \(\gamma\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\), true initial condition \(\theta_0\), linearization \(\mathbb{I}_{\theta_0}\) of \(\mathscr{G}\) at \(\theta_0\)), assume the hypotheses of Nickl’s Bernstein von Mises theorem (Conditions 1 and 2 of Nickl (2024) with \(\gamma\) and \(k\) as in his Theorem 2). Let \(\mathscr{G}_{\rm lin}(\theta)=\mathscr{G}(\theta_0)+\mathbb{I}_{\theta_0}[\theta-\theta_0]\) and let \(\Pi_{\rm lin}(\cdot\mid Z^{(N)})\) be the posterior in the model with \(\mathscr{G}_{\rm lin}\) in place of \(\mathscr{G}\), with the same prior and the same data. Then \(\Pi_{\rm lin}(\cdot\mid Z^{(N)})\) is a Gaussian measure, and \(\mathrm{KL}\big(\Pi(\cdot\mid Z^{(N)})\,\big\|\,\Pi_{\rm lin}(\cdot\mid Z^{(N)})\big)\to0\) in \(P^N_{\theta_0}\) probability.
Source: source/chapters/ext/m2a.tex, label m:thm:prkl.
Realized information gain of the posterior with the efficient information at the truth
Corollary 15.23, page 236 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Under the hypotheses of the Kullback Leibler form of the Bernstein von Mises theorem (Nickl’s Conditions 1 and 2 with \(\gamma\) and \(k\) as in his Theorem 2, with the prior and forward map as there; \(\Pi_{\rm lin}\) the posterior of the model linearized at \(\theta_0\)), let \(J\) be a number of modes, \(\theta^J=(\langle\theta,e_j\rangle)_{j\le J}\) with prior \(\Pi^J\) and prior variances \(\sigma_{0,j}^2\), \(m_N\) and \(V_N\) the posterior mean and covariance of \(\theta^J\), \(I_N=\mathrm{KL}(\Pi^J(\cdot\mid Z^{(N)})\,\|\,\Pi^J)\) the realized information gain, and \(\Gamma_N=\mathrm{KL}(\Pi^J(\cdot\mid Z^{(N)})\,\|\,N(m_N,V_N))\). Then \(\Gamma_N\le\mathrm{KL}\big(\Pi(\cdot\mid Z^{(N)})\,\|\,\Pi_{\rm lin}(\cdot\mid Z^{(N)})\big)\) for every \(J\), so \(\Gamma_N\to0\) in \(P^N_{\theta_0}\) probability uniformly in \(J\). For fixed \(J\), in \(P^N_{\theta_0}\) probability, \(NV_N\to\Sigma_J(\theta_0)=\big(T\langle e_j,(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1}e_{j'}\rangle\big)_{j,j'\le J}\), the covariance of the first \(J\) coordinates under \(\mathcal{N}_{\theta_0}=N(0,T(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1})\), which is positive definite, and \(I_N=\frac12\sum_{j\le J}\big[\log(1+\kappa_j)-\frac{\kappa_j}{1+\kappa_j}\big]+\frac12\,m_N^\top(\Sigma^J_0)^{-1}m_N+o_P(1)\), where the \(\kappa_j\) are the eigenvalues of \(N(\Sigma^J_0)^{1/2}\Sigma_J(\theta_0)^{-1}(\Sigma^J_0)^{1/2}\) and \(\Sigma_0^J=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J}\). For the heat and Stokes equations \(\Sigma_J(\theta_0)=\operatorname{diag}(1/\iota_j)\) and \(\kappa_j=N\iota_j\sigma_{0,j}^2\), where \(\iota_j=\frac1T\int_0^Te^{-2\nu\lambda_jt}\,\mathrm{d}t\).
Source: source/chapters/ext/m2a.tex, label m:cor:prreal.
Entropic van Trees bound
Proposition 15.25, page 238 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\mathscr{G}\) be a measurable forward map in the model \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) (\(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\), \(\varepsilon_i\sim N(0,I_W)\)), let the prior \(\Pi\) be a centered Gaussian measure under which the coordinates \(\langle\theta,e_j\rangle\) are independent with variances \(\sigma_{0,j}^2>0\), and assume \(\int\|\mathscr{G}(\vartheta)\|^2_{L^\infty(\mathcal X)}\,\mathrm{d}\Pi(\vartheta)<\infty\). Fix the number of modes \(J\), let \(\theta^J=(\langle\theta,e_j\rangle)_{j\le J}\), \(\Sigma^J_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J}\), \(I_N=\mathrm{KL}(\Pi^J(\cdot\mid Z^{(N)})\,\|\,\Pi^J)\) the realized information gain, \(\mathbf{E}\) the expectation under the prior predictive law, and \(\mathcal J^J_N(\vartheta)=\frac NT\big(\langle\mathbb{I}_\vartheta e_j,\mathbb{I}_\vartheta e_{j'}\rangle_{L^2(\mathcal X)}\big)_{j,j'\le J}\) the \(J\) block of the Fisher information of \(N\) observations at \(\vartheta\), with \(\mathbb{I}_\vartheta\) the linearization of \(\mathscr{G}\) at \(\vartheta\). Assume that for \(\Pi\) almost every \(\eta\) in the closed span of \((e_j)_{j>J}\) the map \(x\mapsto\mathscr{G}(\eta+\sum_{j\le J}x_je_j)\) from \(\mathbb{R}^J\) into \(C(\mathcal X)\) is continuously differentiable, with partial derivatives \(\mathbb{I}_\vartheta e_j\) at \(\vartheta=\eta+\sum_{j\le J}x_je_j\). If \(\bar{\mathcal J}^J_N=\int\mathcal J^J_N(\vartheta)\,\mathrm{d}\Pi(\vartheta)\) is finite, then for every \(N\), \(\mathbf{E}\,I_N=I(\theta^J;Z^{(N)})\le\frac12\log\det\big(\mathrm{Id}+(\Sigma^J_0)^{1/2}\,\bar{\mathcal J}^J_N\,(\Sigma^J_0)^{1/2}\big)\). If moreover the prior is the rescaled Gaussian series prior with exponent \(\gamma>d\), and the matrix \(\mathcal J^J_N(\vartheta)/N\), which does not depend on \(N\), is continuous at \(\vartheta=0\) for the supremum norm, bounded by a polynomial in \(\|\vartheta\|_\infty\), and invertible at \(\vartheta=0\), then for fixed \(J\), \(\mathbf{E}\,I_N\le\frac12\log\det\big(\mathrm{Id}+(\Sigma^J_0)^{1/2}\,\mathcal J^J_N(0)\,(\Sigma^J_0)^{1/2}\big)+o(1)\).
Source: source/chapters/ext/m2a.tex, label m:prop:prvt.
Reaction diffusion: the expected gain is set at the center of the prior
Proposition 15.26, page 239 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\partial_tu=\Delta u+f(u)\), \(u(0)=\theta\), on the torus \(\Omega=[0,1]^d\) with \(f\in C_c^\infty(\mathbb{R})\) and \(d\le3\), observed as in the model \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) with \(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\) and \(\varepsilon_i\sim N(0,1)\); let \((e_j)\) be the real trigonometric basis, of the functions \(\sqrt2\cos(2\pi k\cdot x)\) and \(\sqrt2\sin(2\pi k\cdot x)\) with eigenvalues \(\lambda_j\); let the prior \(\Pi\) be the rescaled Gaussian series prior with exponent \(\gamma>d\) and variances \(\sigma_{0,j}^2=\lambda_j^{-\gamma}/(N\delta_N^2)\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\); and let the number of modes \(J\) be fixed. Let \(I_N\) be the realized information gain for the first \(J\) modes and \(\mathbf{E}\) the expectation under the prior predictive law. Let \(v\) solve \(v'=f(v)\), \(v(0)=0\), \(\chi(t)=\int_0^tf'(v(s))\,\mathrm{d}s\), \(\iota^0_j=\frac1T\int_0^Te^{2\chi(t)-2\lambda_jt}\,\mathrm{d}t\) and \(\kappa^0_j=N\iota^0_j\sigma_{0,j}^2\) (the information at the center of the prior). Then: (i) \(\mathbf{E}\,I_N\le\frac12\sum_{j\le J}\log(1+\kappa^0_j)+o(1)\). (ii) If \(f\) vanishes on \([-a,a]\) for some \(a>0\), then \(\iota^0_j=\iota_j=(1-e^{-2\lambda_jT})/(2\lambda_jT)\), the value for the heat equation with \(\nu=1\), and \(\mathbf{E}\,I_N=\frac12\sum_{j\le J}\log(1+\kappa^0_j)+o(1)\). (iii) Let \(d\ge2\) and let \(f\) be as in (ii), with moreover \(f\ge0\), \(f\) nondecreasing on \((-\infty,b]\) and strictly increasing on \((a,b)\) for some \(b>a\). The first mode \(e_1\), with \(\lambda_1=4\pi^2\), is \(\sqrt2\cos(2\pi x_p)\) or \(\sqrt2\sin(2\pi x_p)\) for some coordinate \(p\); let \(\theta_0(x)=A\cos(2\pi x_q)\) with \(q\ne p\) and \(a<A<b\), and let \(T\) be so small that the solution of \(w'=f(w)\), \(w(0)=A\), stays at most \(b\) on \([0,T]\). Then \(\|\mathbb{I}_{\theta_0}e_1\|^2_{L^2(\mathcal X)}>T\iota_1\), and for \(J=1\), writing \(\mathcal J^1_N(\theta_0)=\frac NT\|\mathbb{I}_{\theta_0}e_1\|^2_{L^2(\mathcal X)}\) for the Fisher information at \(\theta_0\), \(\frac12\log\big(1+\sigma^2_{0,1}\mathcal J^1_N(\theta_0)\big)-\mathbf{E}\,I_N\to\frac12\log\frac{\|\mathbb{I}_{\theta_0}e_1\|^2_{L^2(\mathcal X)}}{T\iota_1}>0\). (iv) Under the hypotheses of (iii), let \(\psi(r)=(1-e^{-2\lambda_1r})/(2\lambda_1)\), let \(\psi(T-t)f'(u_{\theta_0}(t,x))<2\) for \((t,x)\in\mathcal X\) (as when \(T\sup f'<2\)), and let \(\lambda_j=\lambda_1\) for \(j\le J\). Then \(\Sigma_J(0)-\Sigma_J(\theta_0)\) is positive definite, where \(\Sigma_J(\vartheta)=\big(T\langle e_j,(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}e_{j'}\rangle\big)_{j,j'\le J}\) is the covariance of the Bernstein von Mises limit in the first \(J\) modes and \(\Sigma_J(0)=\mathrm{Id}/\iota_1\), and \(\frac12\log\det\big(\mathrm{Id}+N\sigma^2_{0,1}\Sigma_J(\theta_0)^{-1}\big)-\mathbf{E}\,I_N\to\frac12\log\det\big(\Sigma_J(0)\Sigma_J(\theta_0)^{-1}\big)>0\).
Source: source/chapters/ext/m2a.tex, label m:prop:prrd.
Quadratic variation and the sufficient statistic
Proposition 15.28, page 243 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the linear model \(Y=\Phi\theta+\varepsilon\) with design matrix \(\Phi_{ij}=s_j(t_i)e_j(x_i)\) and unknown noise variance \(\varsigma^2\) (\(\varepsilon\sim N(0,\varsigma^2\mathrm{Id}_N)\)), conditionally on the design the pair \(\big(\Phi^\top Y,\ \|Y\|^2\big)\) is sufficient for \((\theta,\varsigma^2)\), and the conditional law of \(\theta\) given the pair and \(\varsigma^2\) depends on the data only through \(\Phi^\top Y\). If instead the observation noise is replaced by an observed diffusion path on \([0,T]\) with unknown diffusion coefficient, \(\mathrm{d}Y_t=\mathscr{G}(\theta)(t)\,\mathrm{d}t+\sigma(t)\,\mathrm{d}B_t\), the coefficient is identified by the quadratic variation, the likelihood ratio for \(\theta\) is the Girsanov density, and the sufficient statistic for \(\theta\) is again the adjoint evolution driven by the data, now with the data measure replaced by \(\mathrm{d}Y_t/\sigma^2(t)\).
Source: source/chapters/ext/m2a.tex, label m:prop:qv.
The one step summary under four ABC criteria: Bayes risk optimal up to o(1), all four attained in the J mode Gaussian projection
Proposition 15.29, page 244 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on the uniform Bernstein von Mises assumption (Assumption U) and the uniform moment bound (MB).
Statement. Work in Nickl’s framework (torus \(\Omega=[0,1]^d\), data \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\) with \(X_i\) i.i.d. uniform on \(\mathcal X=[0,T]\times\Omega\) and \(\varepsilon_i\sim N(0,I_W)\), Laplace eigenbasis \((e_j)\) with eigenvalues \(\lambda_j\), rescaled Gaussian series prior \(\Pi\) with exponent \(\gamma\) and variances \(\sigma_{0,j}^2\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\), \(J_N\asymp N\delta_N^2\), linearization \(\mathbb{I}_\vartheta\) of \(\mathscr{G}\), \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1})\), \(\tilde\theta_N\) the posterior mean). Assume the hypotheses of Nickl’s Bernstein von Mises theorem (Conditions 1 and 2 of Nickl (2024) with \(\gamma\) and \(k\) as in his Theorem 2) and the hypotheses of the equivalence of the cross fitted one step efficient score summary \(\hat S_N=\hat S^{J_N}_N\) with the efficient centering (catalog entry equivalence_with_the_efficient_centering); assume the uniform Bernstein von Mises statement (Assumption U) both with \(S_N=\hat S_N\) and with \(S_N=\tilde\theta_N\); and assume the uniform moment bound (MB): for some \(\varpi>0\), \(\sup_N\mathbf{E}\,\|\sqrt N(\theta-\hat S_N)\|^{2+\varpi}_{H^{-k-2}}+\sup_N\mathbf{E}\,\|\sqrt N(\theta-\tilde\theta_N)\|^{2+\varpi}_{H^{-k-2}}<\infty\), the expectations being under the prior predictive law \(\mathbf{P}=\Pi\otimes P^N_\vartheta\) (parameter \(\vartheta\sim\Pi\), then data \(Z^{(N)}\sim P^N_\vartheta\)).
Let \(\mathcal S_J\) be the class of all measurable statistics \(S=g(Z^{(N)})\) taking values in a \(J\) dimensional subspace of \(H^{-k-2}\), with \(J=J_N\); let \(P_J\) be the projection onto \(\operatorname{span}\{e_j:j\le J\}\), \(\theta^J=P_J\theta\), and \(\mathscr{W}_1=\mathscr{W}_{1,H^{-k-2}}\). In the first \(J\) modes put \(\Sigma_J(\vartheta)=\big(T\langle e_j,(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1}e_{j'}\rangle\big)_{j,j'\le J}\) (the covariance of \(\mathcal{N}_\vartheta\)), \(\mathcal J^J_N(\vartheta)=\frac NT\big(\langle e_j,\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta e_{j'}\rangle\big)_{j,j'\le J}\) (the \(J\) block of the Fisher information of \(N\) observations), \(\Sigma^J_0=\operatorname{diag}(\sigma_{0,j}^2)_{j\le J}\) and \(V^J_N(\vartheta)=\big(\mathcal J^J_N(\vartheta)+(\Sigma^J_0)^{-1}\big)^{-1}\). For \(S\in\mathcal S_J\) consider four criteria: (i) the expected posterior entropy \(\mathcal H_N(S)=\mathbf{E}\,H\big(\operatorname{Law}(\theta^J\mid S)\big)\), to be minimized; (ii) \(\mathcal F_N(S)=\log\det\mathcal J_N(S)\), where \(\mathcal J_N(S)\) is the efficient Fisher information carried at \(\theta_0\) by the induced experiment \(\{\operatorname{Law}_\theta(S)\}\) about the \(J\) mode subparameter \((\langle e_j,\theta\rangle)_{j\le J}\), to be maximized; (iii) the \(L^2\) Bayes risk \(\mathcal R_N(S)=N\,\mathbf{E}\,\|\theta-\mathbf{E}[\theta\mid S]\|^2_{H^{-k-2}}\), to be minimized; (iv) the expected information gain \(\mathcal U_N(S)=I(\theta^J;S)\) (mutual information), to be maximized.
Then \(\hat S_N\in\mathcal S_J\), and: (a) (exact at every \(N\)) for every \(S\in\mathcal S_J\), \(\mathcal H_N(S)\ge\mathcal H_N(Z^{(N)})\), \(\mathcal F_N(S)\le\mathcal F_N(Z^{(N)})\), \(\mathcal R_N(S)\ge\mathcal R_N(Z^{(N)})\) and \(\mathcal U_N(S)\le\mathcal U_N(Z^{(N)})\); moreover (i) and (iv) are one criterion, since \(\mathcal U_N(S)=H\big(\operatorname{Law}(\theta^J)\big)-\mathcal H_N(S)\) for every \(N\) and every \(S\), so entropy minimization and information gain maximization have identical optimizers exactly. (b) Under (MB), \(\mathcal R_N(\hat S_N)-\mathcal R_N(Z^{(N)})\to0\) and both terms are bounded in \(N\). If moreover the weighted traces \(t(\vartheta)=T\sum_{j\ge1}(1+\lambda_j)^{-k-2}\langle e_j,(\mathbb{I}_\vartheta^*\mathbb{I}_\vartheta)^{-1}e_j\rangle\) of the limits \(\mathcal{N}_\vartheta\) satisfy \(\sup_N\int t^{1+\varpi/2}\,\mathrm{d}\Pi<\infty\), as they do for the heat and Stokes equations, where \(t\) is constant, then both terms equal \(\varrho_N+o(1)\) with \(\varrho_N=\int t(\vartheta)\,\mathrm{d}\Pi(\vartheta)\), which depends on \(N\) only through the rescaled prior \(\Pi\) and is bounded; so \(\hat S_N\) attains the criterion (iii) optimum within \(\mathcal S_J\) up to \(o(1)\). (c) Call the \(J\) mode Gaussian projection at \(\vartheta\) the linear Gaussian experiment with parameter \(\theta^J\in\mathbb{R}^J\), prior \(N(0,\Sigma^J_0)\) and information matrix \(\mathcal J^J_N(\vartheta)\), in which the modes beyond \(J\) are known; criteria (i) and (iv) are evaluated in it at the parameter \(\vartheta\sim\Pi\) that generates the data, and criterion (ii) at \(\vartheta=\theta_0\). In it the regularized form of \(\hat S_N\) is exactly sufficient, so \(\operatorname{Law}(\theta^J\mid\hat S_N)=\operatorname{Law}(\theta^J\mid Z^{(N)})=N(\cdot,V^J_N(\vartheta))\) there, and all four criteria take identical values at \(\hat S_N\) and at the full data, namely \(\mathcal H^\star_N=\frac12\int\log\det\big(2\pi e\,V^J_N(\vartheta)\big)\,\mathrm{d}\Pi(\vartheta)\), \(\mathcal U^\star_N=\frac12\int\log\det\big(\Sigma^J_0\,V^J_N(\vartheta)^{-1}\big)\,\mathrm{d}\Pi(\vartheta)\), which equals \(\frac12\sum_{j\le J}\log(1+\kappa_j)\) with \(\kappa_j=N\iota_j\sigma_{0,j}^2\) for the heat and Stokes equations (where \(V^J_N(\vartheta)\) does not depend on \(\vartheta\) and is diagonal), and \(\mathcal F^\star_N=\log\det\mathcal J^J_N(\theta_0)\). For a nonlinear forward map the integral defining \(\mathcal U^\star_N\) need not equal its value at \(\theta_0\). In the exact experiment, where the modes beyond \(J\) are unknown parameters, the efficient information for the first \(J\) modes is the Schur complement \(N\Sigma_J(\theta_0)^{-1}\preceq\mathcal J^J_N(\theta_0)\), and \(\hat S_N\) attains \(\log\det\big(N\Sigma_J(\theta_0)^{-1}\big)\) asymptotically, for each fixed \(J\); the two values of criterion (ii) agree when \(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) is diagonal in \((e_j)\). For the heat and Stokes equations the \(J\) mode Gaussian projection does not depend on \(\vartheta\) and is the limit of the exact experiment at the \(\sqrt N\) scale in \(\mathscr{W}_1\); when \(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) couples the first \(J\) modes with the others it is not that limit.
For criteria (i), (ii) and (iv), part (c) is a statement about the \(J\) mode Gaussian projection and not about the exact finite \(N\) functionals: the book does not claim, and the hypotheses do not give, \(\mathcal H_N(\hat S_N)-\mathcal H_N(Z^{(N)})\to0\) or \(\mathcal F_N(\hat S_N)-\mathcal F_N(Z^{(N)})\to0\) for the exact laws, because differential entropy, Kullback Leibler divergence and Fisher information are not continuous along \(\mathscr{W}_1\) convergence. Criterion (iii) is the exception, being a second moment functional.
Source: source/chapters/ext/m2b.tex, label m:prop:abcopt.
The adjoint statistic is the PLS projection
Proposition 15.32, page 252 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider the linear model \(Y=\Phi\theta+\varepsilon\) with design matrix \(\Phi_{ij}=s_j(t_i)e_j(x_i)\) (\(i\le N\), \(j\le D\)), prior \(\theta\sim N(0,\Sigma_0)\) and noise \(\varepsilon\sim N(0,\mathrm{Id}_N)\) independent of \(\theta\), and treat the raw data \(Y\in\mathbb{R}^N\) as the pool of candidate statistics. The first partial least squares (PLS) direction for predicting a coordinate \(\theta_j\) from \(Y\) is the vector \(w\in\mathbb{R}^N\) maximizing \(\mathrm{Cov}(\theta_j,w^\top Y)\) subject to \(\|w\|=1\), and \(\mathrm{Cov}(\theta_j,w^\top Y)=w^\top\mathrm{Cov}(Y,\theta_j)=w^\top\Phi\,\Sigma_0\mathbf{u}_j\), where \(\mathbf{u}_j\) is the \(j\)th unit vector of \(\mathbb{R}^D\); so the direction is \(\Phi\Sigma_0\mathbf{u}_j/\|\Phi\Sigma_0\mathbf{u}_j\|\) and the corresponding score is proportional to \((\Sigma_0\Phi^\top Y)_j\). Collecting \(j\le D\) and absorbing the fixed prior covariance, the PLS scores are exactly the coordinates of the adjoint statistic \(\Phi^\top Y\) (the coefficient vector of the time zero value of the backward adjoint solution driven by the data), which is sufficient. In particular a single PLS component per parameter coordinate suffices, and no further components add information.
Source: source/chapters/ext/m2b.tex, label m:prop:pls.
Deep PLS structure of the GBC summary
Proposition 15.34, page 253 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on the uniform Bernstein von Mises assumption (Assumption U), for the posterior given the scores alone.
Statement. Let \(\mathscr{G}\) be a nonlinear forward map in Nickl’s framework and let the summary be the cross fitted one step efficient score summary \(\hat S_N=\hat S^{J_N}_N\) (catalog entry equivalence_with_the_efficient_centering), whose coordinates average over the two folds the preliminary estimate coordinate \(\langle e_j,\bar\theta^{[b]}_N\rangle\) plus the average, over the other half sample, of the residual \(Y_i-u_{\bar\theta^{[b]}_N}(X_i)\) projected on the direction \(\mathbb{I}_{\bar\theta^{[b]}_N}p_N(\bar e_j^{\,\bar\theta^{[b]}_N})(X_i)\). Then: (i) the first stage is a linear projection of the residuals onto the directions \(\mathbb{I}_{\bar\theta_N}p_N(\bar e_j)\), that is, a partial least squares step with directions supplied by the linearized forward map rather than estimated from the simulations; (ii) the second stage, the map from the projected scores \(s\) to the posterior, is asymptotically the identity plus a scale, namely the conditional quantile map satisfies \(Q_j(\tau\mid s)=s_j+z_\tau\sigma_j/\sqrt N+o(N^{-1/2})\), with \(z_\tau\) the \(\tau\) quantile of the standard normal law and \(\sigma_j^2=T\langle e_j,(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1}e_j\rangle\), read in the Wasserstein sense, under the hypotheses of the equivalence of \(\hat S_N\) with the efficient centering for the posterior given all of the data, and under Assumption U as well for the posterior given the scores alone; so under these hypotheses the deep part of deep PLS has an asymptotically trivial link.
Source: source/chapters/ext/m2b.tex, label m:prop:dpls.
Jiang Wong under a uniform Bernstein von Mises assumption: the posterior mean summary is asymptotically sufficient
Theorem 15.38, page 256 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on the uniform Bernstein von Mises assumption (Assumption U).
Statement. Let \(Z^{(N)}=(Y_i,X_i)_{i\le N}\) with \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\), \(X_i\) i.i.d. uniform on \([0,T]\times\Omega\) and independent \(\varepsilon_i\sim N(0,I_W)\), where \(\mathscr{G}(\theta)=u_\theta\) is the solution map of an evolution equation with initial condition \(\theta\). Let the prior \(\Pi\) be the law of \((\sqrt N\delta_N)^{-1}\sum_j\lambda_j^{-\gamma/2}g_je_j\), with \((e_j,\lambda_j)\) the Laplace eigenpairs, \(g_j\) i.i.d. \(N(0,1)\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\), and let the prior and the forward map satisfy Conditions 1 and 2 of Nickl (2024) with the constants of his Bernstein von Mises theorem (integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\), where \(\zeta>d/2\) and \(\eta_0\) are the indices of his Condition 2), so that, with \(\tilde\theta_N=E^\Pi[\theta\mid Z^{(N)}]\) the posterior mean, \(\mathbb{I}_\vartheta\) the linearization of \(\mathscr{G}\) at \(\vartheta\), \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\) and \(\mathscr{W}_1\) the Wasserstein distance of order one on the negative Sobolev space \(H^{-k-2}\), \(\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-\tilde\theta_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\to0\) in \(P^N_{\theta_0}\) probability, \(\theta_0\) being the true initial condition and \(P^N_\vartheta\) the law of the data under \(\vartheta\). Let \(\mathbf{P}=\Pi\otimes P^N_\vartheta\) be the joint law of a parameter \(\vartheta\) drawn from the prior and data drawn given it, which is also the law under which the GBC training pairs are simulated.
Assume the uniform Bernstein von Mises assumption for the summary \(S_N=\tilde\theta_N\): there are sets \(\Theta_N\) such that (U1) \(\Pi(\Theta_N)\to1\), \(\theta_0\in\Theta_N\) for all large \(N\), and \(\sup_{\vartheta\in\Theta_N}E^N_\vartheta\,\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_\vartheta\big)\to0\); (U2) these discrepancies are uniformly integrable under the prior predictive law; (U3) \(\vartheta\mapsto T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}\), into bounded operators \(H^{-k-2}\to H^{-k-4}\), and \(\vartheta\mapsto\mathcal{N}_\vartheta\), into the probability measures on \(H^{-k-2}\) with the distance \(\mathscr{W}_1\), are Lipschitz on \((\Theta_N,\|\cdot\|_{H^\zeta})\) with constants that do not depend on \(N\), the first moments \(m_1(\vartheta)=\int\|x\|_{H^{-k-2}}\,\mathrm{d}\mathcal{N}_\vartheta(x)\) satisfy \(\int_{\Theta_N^c}m_1\,\mathrm{d}\Pi\to0\) and \(\Pi(\Theta_N^c)\sup_{\Theta_N}m_1\to0\), \(S_N\in\Theta_N\) with probability tending to one, and \(E^\Pi[\|\theta-S_N\|_{H^\zeta}\mid S_N]\to0\); (U4) for every \(\theta_0\in\Theta_N\), \(E\big[\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\mid S_N\big]\to0\) in \(P^N_{\theta_0}\) probability.
Let \(\tau_N=\operatorname{Law}(\sqrt N(\theta-\tilde\theta_N)\mid Z^{(N)})\) be the exact recentered posterior and \(\tau^{\rm S}_N=\operatorname{Law}(\sqrt N(\theta-\tilde\theta_N)\mid\tilde\theta_N)\) the recentered posterior given the posterior mean statistic alone, both conditional laws under \(\mathbf{P}\). Then \(\mathscr{W}_1(\tau^{\rm S}_N,\mathcal{N}_{\tilde\theta_N})\to0\) and \(\mathscr{W}_1(\tau_N,\tau^{\rm S}_N)\to0\) in \(\mathbf{P}\) probability, and, because \(\theta_0\in\Theta_N\), both also hold in \(P^N_{\theta_0}\) probability with \(\mathcal{N}_{\tilde\theta_N}\) replaced by \(\mathcal{N}_{\theta_0}\). The same holds with \(\tilde\theta_N\) replaced by the cross fitted one step efficient score statistic \(\hat S^{J_N}_N\) (catalog entry equivalence_with_the_efficient_centering) when the assumption above holds with \(S_N=\hat S^{J_N}_N\) and the hypotheses of that entry hold. In words: whenever a Bernstein von Mises theorem holds uniformly over prior typical truths, the posterior mean summary, or any summary \(\sqrt N\) equivalent to it in \(H^{-k-2}\), is asymptotically sufficient in Wasserstein distance of order one at the \(\sqrt N\) scale.
Source: source/chapters/ext/m2c.tex, label m:thm:jwbvm.
GBC with the posterior mean summary
Corollary 15.40, page 258 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on the uniform Bernstein von Mises assumption (Assumption U), for the root N statement only.
Statement. Let data \(Z^{(N)}\) come from a forward map and rescaled Gaussian series prior \(\Pi\) that satisfy Conditions 1 and 2 of Nickl (2024) with the constants of his Bernstein von Mises theorem, as in the catalog entry jiang_wong_under_bernstein_von_mises, where \(T\) is the length of the observation window, \(\mathbb{I}_\vartheta\) the linearization of the forward map at \(\vartheta\), and \(\zeta>d/2\) and \(k\) the Sobolev indices of that theorem; let \(\Pi_N=\Pi(\cdot\mid Z^{(N)})\) be the posterior, \(\tilde\theta_N\) its mean, \(V(Z^{(N)})=\mathrm{Var}(\theta\mid Z^{(N)})\) its covariance, \(\theta_0\) the true parameter, and \(\mathscr{W}_1\) the Wasserstein distance of order one on the negative Sobolev space \(H^{-k-2}\). Let \(\Pi^{\rm S}_N=\operatorname{Law}(\theta\mid\tilde\theta_N)\), the conditional law, under the joint law of prior and data, of the parameter given the posterior mean statistic alone; it is the target of generative Bayesian computation (GBC) with the posterior mean summary and an exact generator. Then for every \(N\) the mean of \(\Pi^{\rm S}_N\) equals the posterior mean \(\tilde\theta_N\) and its covariance is \(E[V(Z^{(N)})\mid\tilde\theta_N]\). If moreover the uniform Bernstein von Mises assumption holds with \(S_N=\tilde\theta_N\) (sets \(\Theta_N\) of prior mass tending to one that contain \(\theta_0\), on which the recentered posterior is uniformly close to its Gaussian limit \(N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\), with the integrability, Lipschitz, moment and conditional clauses (U1) to (U4) stated in that entry), then \(\sqrt N\,\mathscr{W}_1(\Pi^{\rm S}_N,\Pi_N)\to0\) in \(P^N_{\theta_0}\) probability, and also in probability under the joint law of prior and data; this is the accuracy condition (G1) that the book’s transfer theorem places on a GBC posterior.
Source: source/chapters/ext/m2c.tex, label m:cor:jwgbc.
Sufficiency gap in the linear Gaussian model: Rao Blackwell lower bound and transport entropy upper bound
Theorem 15.41, page 258 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the linear Gaussian model \(Y=\Phi\theta+\varepsilon\) with \(\theta\sim N(0,\Sigma_0)\) in \(\mathbb{R}^D\) and \(\varepsilon\sim N(0,\varsigma^2\mathrm{Id})\) independent of \(\theta\), let \(Z=Y\) be the full data and \(S=AY\) any linear statistic. Then the posteriors \(\Pi(\cdot\mid Z)\) and \(\Pi(\cdot\mid S)\) are Gaussian with covariances \(V_Z=\mathrm{Var}(\theta\mid Z)\preceq V_S=\mathrm{Var}(\theta\mid S)\) that do not depend on the data. Let \(\mathscr{W}_1\) be the Wasserstein distance of order one on \(\mathbb{R}^D\) with the Euclidean norm and \(\Delta I=I(\theta;Z)-I(\theta;S)\ge0\) the mutual information deficit. Then: (i) (lower bound, Rao Blackwell) \(\mathscr{W}_1(\Pi(\cdot\mid S),\Pi(\cdot\mid Z))\ge\|\xi\|\), where \(\xi=E[\theta\mid S]-E[\theta\mid Z]\) is a centered Gaussian vector with covariance \(V_S-V_Z\), so that \(E\|\xi\|^2=\mathrm{tr}(V_S-V_Z)\) and \(\sqrt{\mathrm{tr}(V_S-V_Z)-\|V_S-V_Z\|_{op}}\le E\|\xi\|\le\sqrt{\mathrm{tr}(V_S-V_Z)}\); (ii) (upper bound, transport entropy) \(E_Z\,\mathscr{W}_1(\Pi(\cdot\mid Z),\Pi(\cdot\mid S))\le\sqrt{2\|V_S\|_{op}\,\Delta I}\), where \(\Delta I=\tfrac12\log\det(V_SV_Z^{-1})\); (iii) the lower bound \(\|\xi\|\) of (i) vanishes almost surely, and the upper bound of (ii) vanishes, if and only if \(V_S=V_Z\), if and only if \(S\) is Bayes sufficient, that is, \(\operatorname{Law}(\theta\mid S)=\operatorname{Law}(\theta\mid Z)\) almost surely; for the adjoint statistic \(S=\Phi^\top Y\) both vanish identically. Consequently, for any GBC posterior \(\hat\Pi\) built on \(S\), with generator error \(\epsilon_{\rm gen}=\mathscr{W}_1(\hat\Pi,\Pi(\cdot\mid S))\), in expectation over the data \(E\,\mathscr{W}_1(\hat\Pi,\Pi(\cdot\mid Z))\ge\sqrt{\mathrm{tr}(V_S-V_Z)-\|V_S-V_Z\|_{op}}-E\,\epsilon_{\rm gen}\) and \(E\,\mathscr{W}_1(\hat\Pi,\Pi(\cdot\mid Z))\le E\,\epsilon_{\rm gen}+\sqrt{2\|V_S\|_{op}\,\Delta I}\).
Source: source/chapters/ext/m2c.tex, label m:thm:gap.
Sufficiency gap beyond the linear model
Proposition 15.42, page 260 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let a parameter \(\theta\), expanded in the Laplace eigenbasis \((e_j)\) with eigenvalues \(\lambda_j\), have a prior \(\Pi\), let \(Z^{(N)}\) be the data and \(\mathbf P\) the joint law of parameter and data, and let \(k\) be the index of the negative Sobolev space \(H^{-k-2}\) in which the posterior is compared with its Gaussian limit. Let \(S=S(Z^{(N)})\) be a statistic, fix \(D\), and write \(\mathscr{W}_{1,w}\) for the Wasserstein distance of order one between laws of the first \(D\) coordinates \((\langle e_j,\theta\rangle)_{j\le D}\) in the norm \(\|x\|_w^2=\sum_{j\le D}(1+\lambda_j)^{-k-2}x_j^2\) that \(H^{-k-2}\) induces on them. Suppose that there are a linear Gaussian model \(Y_G=\Phi_G\theta+\varepsilon_G\) on \(\mathbb{R}^D\), with prior \(N(0,\Sigma_0)\), \(\varepsilon_G\sim N(0,\mathrm{Id})\) and posterior \(\Pi_G\), a linear statistic \(S_G=AY_G\), and a coupling of \((Z^{(N)},S)\) under \(\mathbf P\) with \((Y_G,S_G)\) under the Gaussian model, such that \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid Z^{(N)}),\Pi_G(\cdot\mid Y_G)\big)\to0\) and \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid S),\Pi_G(\cdot\mid S_G)\big)\to0\). Let \(V_Z\preceq V_S\) be the posterior covariances of the Gaussian model given \(Y_G\) and given \(S_G\), \(\Delta I=\tfrac12\log\det(V_SV_Z^{-1})\), and take traces and operator norms in \(\|\cdot\|_w\), that is, \(\mathrm{tr}_wV=\mathrm{tr}(\Lambda_wV\Lambda_w)\) and \(\|V\|_{op,w}=\|\Lambda_wV\Lambda_w\|_{op}\) with \(\Lambda_w=\operatorname{diag}((1+\lambda_j)^{-(k+2)/2})\). Then the sufficiency gap at the \(\sqrt N\) scale, \(\Upsilon_N=\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid S),\Pi(\cdot\mid Z^{(N)})\big)\), satisfies \(\Upsilon_N\ge\sqrt{N\,\mathrm{tr}_w(V_S-V_Z)-N\|V_S-V_Z\|_{op,w}}-o(1)\) and \(\Upsilon_N\le\sqrt{2N\|V_S\|_{op,w}\,\Delta I}+o(1)\). For the heat and Stokes equations with viscosity \(\nu\) and observation window \([0,T]\), in the Gaussian experiment that observes \(\theta_j+(N\iota_j)^{-1/2}\xi_j\), \(j\le D\), with independent standard normal \(\xi_j\), independent priors \(\theta_j\sim N(0,\sigma^2_{0,j})\), per mode information \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\) and \(\kappa_j=N\iota_j\sigma^2_{0,j}\), and with \(S\) its first \(J'\) coordinates, the hypotheses hold with no error, \(N(V_S-V_Z)\) is diagonal with entries \(0\) for \(j\le J'\) and \(N\sigma^2_{0,j}\kappa_j/(1+\kappa_j)\) for \(J'<j\le D\), and \(\Delta I=\tfrac12\sum_{J'<j\le D}\log(1+\kappa_j)\); with \(a_j=(1+\lambda_j)^{-k-2}N\sigma^2_{0,j}\kappa_j/(1+\kappa_j)\) the lower bound is \(\big(\sum_{J'<j\le D}a_j-\max_{J'<j\le D}a_j\big)^{1/2}\).
Source: source/chapters/ext/m2c.tex, label m:prop:gapnl.
The Gaussian comparison models admitted by the first hypothesis of the nonlinear sufficiency gap bounds
Proposition 15.43, page 261 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on clauses (U1) and (U2) of the uniform Bernstein von Mises assumption, and (U3) for part (ii).
Statement. Let data \(Z^{(N)}\) come from the forward map and observation model of Nickl’s (2024) Bernstein von Mises theorem (catalog entry jiang_wong_under_bernstein_von_mises), and let \(\Pi\) be a prior, which may depend on \(N\), as the rescaled Gaussian series prior of that theorem does, or not, let \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\) be the Gaussian limit at the parameter \(\vartheta\), with \(T\) the length of the observation window and \(\mathbb{I}_\vartheta\) the linearization of the forward map at \(\vartheta\), on the negative Sobolev space \(H^{-k-2}\) (\(\zeta>d/2\) and \(k\) the Sobolev indices of Nickl’s theorem), and let \(\mathbf P\) be the joint law of a parameter drawn from the prior and the data. With \((e_j,\lambda_j)\) the Laplace eigenpairs, fix \(D\), let \(\Lambda_w=\operatorname{diag}((1+\lambda_j)^{-(k+2)/2})_{j\le D}\), let \(\mathscr{W}_{1,w}\) be the Wasserstein distance of order one for laws of the first \(D\) coordinates in the norm \(\|x\|_w=|\Lambda_wx|\), and let \(\Sigma_D(\vartheta)\) be the covariance matrix of the first \(D\) coordinates under \(\mathcal{N}_\vartheta\). For symmetric positive semidefinite \(D\times D\) matrices \(V,V'\) and a unit vector \(u\in\mathbb{R}^D\) put \(\sigma_u(V)=(u^\top\Lambda_wV\Lambda_wu)^{1/2}\) and \(\mathrm{dist}_w(V,V')=\max_{|u|=1}|\sigma_u(V)-\sigma_u(V')|\). Consider a linear Gaussian model \(Y_G=\Phi_G\theta+\varepsilon_G\) on \(\mathbb{R}^D\) with prior \(N(0,\Sigma_0)\), \(\varepsilon_G\sim N(0,\mathrm{Id})\), posterior \(\Pi_G\) and posterior covariance \(V_Z\) given \(Y_G\), with a coupling of \(Z^{(N)}\) under \(\mathbf P\) with \(Y_G\); its first hypothesis in the nonlinear sufficiency gap bounds (catalog entry sufficiency_gap_beyond_the_linear_model) is \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid Z^{(N)}),\Pi_G(\cdot\mid Y_G)\big)\to0\).
Let some summary \(S_N\) satisfy clauses (U1) and (U2) of the uniform Bernstein von Mises assumption: there are sets \(\Theta_N\) with \(\Pi(\Theta_N)\to1\) and \(\theta_0\in\Theta_N\) for all large \(N\) such that \(\sup_{\vartheta\in\Theta_N}E^N_\vartheta\,\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_\vartheta\big)\to0\), and these discrepancies are uniformly integrable under the prior predictive law. Let \(\vartheta\mapsto\Sigma_D(\vartheta)\) be measurable. Then for every such linear Gaussian model and every coupling, \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid Z^{(N)}),\Pi_G(\cdot\mid Y_G)\big)\ge\sqrt{2/\pi}\int\mathrm{dist}_w\big(\Sigma_D(\vartheta),NV_Z\big)\,\mathrm{d}\Pi(\vartheta)-o(1)\), with an \(o(1)\) that depends neither on the model nor on the coupling. Consequently: (i) if the prior does not depend on \(N\), the first hypothesis holds for some Gaussian model only if \(\Sigma_D\) is \(\Pi\) almost surely constant; (ii) if \(E^\Pi\|\theta\|_{H^\zeta}\to0\), as under the rescaled prior with \(\gamma\) as in Nickl’s theorem, and clause (U3) holds with \(0\in\Theta_N\) for all \(N\) (Lipschitz dependence of \(T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}\) and of \(\mathcal{N}_\vartheta\) on \(\vartheta\in\Theta_N\) in the \(H^\zeta\) norm, with the moment and localization clauses stated in the catalog entry jiang_wong_under_bernstein_von_mises), then \(\int\mathrm{dist}_w\big(\Sigma_D(\vartheta),\Sigma_D(0)\big)\,\mathrm{d}\Pi(\vartheta)\to0\), and the first hypothesis forces \(NV_Z\to\Sigma_D(0)\), the inverse information at the prior mean read in the first \(D\) coordinates; in particular a Gaussian model with \(NV_Z\to\Sigma_D(\theta_0)\), the covariance of the Bernstein von Mises limit at the truth, satisfies it only if \(\Sigma_D(\theta_0)=\Sigma_D(0)\).
Source: source/chapters/ext/m2c.tex, label m:prop:gapreq.
A sufficient condition for the nonlinear sufficiency gap hypotheses in terms of the centering
Lemma 15.44, page 263 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on clauses (U1) to (U3) of the uniform Bernstein von Mises assumption.
Statement. Let data \(Z^{(N)}\), a prior \(\Pi\), the joint law \(\mathbf P\) of parameter and data, the Gaussian limits \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\) on the negative Sobolev space \(H^{-k-2}\), the weighted norm \(\|x\|_w^2=\sum_{j\le D}(1+\lambda_j)^{-k-2}x_j^2\) on the first \(D\) coordinates in the Laplace eigenbasis, the distance \(\mathscr{W}_{1,w}\), and the covariance \(\Sigma_D(\vartheta)\) of the first \(D\) coordinates under \(\mathcal{N}_\vartheta\) be as in the catalog entry sufficiency_gap_admissible_gaussian_models, and let \(P_D\) map \(\theta\) to its first \(D\) coordinates. Let a summary \(S_N\) satisfy clauses (U1) to (U3) of the uniform Bernstein von Mises assumption (stated in the catalog entry jiang_wong_under_bernstein_von_mises) with \(0\in\Theta_N\) for all \(N\), and let \(E^\Pi\|\theta\|_{H^\zeta}\to0\). Then for every statistic \(S=S(Z^{(N)})\), \(\mathbf E\,\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N\theta\mid S),\operatorname{Law}(\sqrt NS_N\mid S)*\mathcal{N}_0\big)\to0\), where \(*\) is convolution and \(\mathcal{N}_0\) is the Gaussian limit at the prior mean \(0\). Let a linear Gaussian model \(Y_G=\Phi_G\theta_G+\varepsilon_G\) on \(\mathbb{R}^D\), with prior \(N(0,\Sigma_0)\), \(\varepsilon_G\sim N(0,\mathrm{Id})\) and posterior covariance \(V_Z\) given \(Y_G\), satisfy \(NV_Z\to\Sigma_D(0)\), write \(m_G=E[\theta_G\mid Y_G]\), let \(S_G=AY_G\) be a linear statistic, and let a coupling of \(S\) under \(\mathbf P\) with \(S_G\) under the Gaussian model satisfy \(E\,\mathscr{W}_{1,w}\big(\operatorname{Law}(\sqrt NP_DS_N\mid S),\operatorname{Law}(\sqrt Nm_G\mid S_G)\big)\to0\). Then it extends to a coupling of \((Z^{(N)},S)\) with \((Y_G,S_G)\) under which both hypotheses of the nonlinear sufficiency gap bounds hold: \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid Z^{(N)}),\Pi_G(\cdot\mid Y_G)\big)\to0\) and \(\sqrt N\,E\,\mathscr{W}_{1,w}\big(\Pi(\cdot\mid S),\Pi_G(\cdot\mid S_G)\big)\to0\).
Source: source/chapters/ext/m2c.tex, label m:lem:gapred.
A two coordinate model in which the sufficiency gap hypotheses depend on the summary
Proposition 15.45, page 264 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: counterexample, proved.
Statement. Let \(\theta\in\mathbb{R}^2\) have the prior \(N(0,s_N^2\mathrm{Id})\) with \(s_N\to0\) and \(Ns_N^2\to\infty\), and let \(Y_i=G(\theta)+\varepsilon_i\), \(i\le N\), with independent \(\varepsilon_i\sim N(0,\mathrm{Id})\) and \(G(\theta)=(\theta_1+\theta_2^2,\theta_2)\). The mean \(X\) of the \(Y_i\) is sufficient, \(DG(\theta)\) has rows \((1,2\theta_2)\) and \((0,1)\), and \((DG(\theta)^\top DG(\theta))^{-1}\) has first diagonal entry \(1+4\theta_2^2\). Let \(\mathscr{W}_1\) be the Wasserstein distance of order one on \(\mathbb{R}^2\) with the Euclidean norm and \(\mathrm{dist}(V,V')=\max_{|u|=1}|(u^\top Vu)^{1/2}-(u^\top V'u)^{1/2}|\). A linear Gaussian model on \(\mathbb{R}^2\) is \(Y_G=\Phi_G\theta_G+\varepsilon_G\) with a prior \(N(0,\Sigma_0)\) and \(\varepsilon_G\sim N(0,\mathrm{Id})\), with posterior \(\Pi_G\) and posterior covariance \(V_Z\) given \(Y_G\); for \(\theta_\ast\in\mathbb{R}^2\) the Gaussian model at \(\theta_\ast\) is \(Y_G=\sqrt N\,DG(\theta_\ast)\theta_G+\varepsilon_G\) with the prior \(N(0,s_N^2\mathrm{Id})\). For a statistic \(S\) of \(X\), a linear statistic \(S_G\) of \(Y_G\) and a coupling, the first hypothesis of the nonlinear sufficiency gap bounds is \(\sqrt N\,E\,\mathscr{W}_1(\Pi(\cdot\mid X),\Pi_G(\cdot\mid Y_G))\to0\) and the second is \(\sqrt N\,E\,\mathscr{W}_1(\Pi(\cdot\mid S),\Pi_G(\cdot\mid S_G))\to0\); those bounds hold in this setting with the same proof. Then:
coupled through \(\theta_G=\theta\) and \(\varepsilon_G=\sqrt N(X-G(\theta))\), the Gaussian model at the prior mean \(\theta_\ast=0\) satisfies the first hypothesis;
for every linear Gaussian model on \(\mathbb{R}^2\) and every coupling, \(\liminf_N\sqrt N\,E\,\mathscr{W}_1(\Pi(\cdot\mid X),\Pi_G(\cdot\mid Y_G))\ge\sqrt{2/\pi}\,\liminf_N\mathrm{dist}(NV_Z,\mathrm{Id})\); for a point \(\theta_0\in\mathbb{R}^2\) the Gaussian model at \(\theta_\ast=\theta_0\) has \(NV_Z\to(DG(\theta_0)^\top DG(\theta_0))^{-1}\), so it fails the first hypothesis whenever \(\theta_{0,2}\ne0\), by at least \(\sqrt{2/\pi}\,((1+4\theta_{0,2}^2)^{1/2}-1)\);
let \(S=X_1\), the first coordinate of the adjoint statistic \(DG(0)^\top X\) at the prior mean, which is also the one step summary \(\bar\theta+DG(\bar\theta)^{-1}(X-G(\bar\theta))\) from \(\bar\theta=0\); if \(\sqrt Ns_N^2\to\infty\), then for every linear Gaussian model on \(\mathbb{R}^2\), every linear statistic \(S_G\) and every coupling, \(\sqrt N\,E\,\mathscr{W}_1(\Pi(\cdot\mid S),\Pi_G(\cdot\mid S_G))\to\infty\), and its ratio to \(\sqrt Ns_N^2\) has \(\liminf\) at least \(c_\ast=\sup_{0<\tau\le1/2}\kappa_\tau/(2(1-\tau))\), where \(\kappa_\tau=\big|\sqrt{2x_\tau/\pi}\,e^{-x_\tau/2}-\sqrt{2x_{1-\tau}/\pi}\,e^{-x_{1-\tau}/2}\big|\) and \(x_\tau\) is the \(\tau\) quantile of the \(\chi^2_1\) law (numerically \(c_\ast=0.1588\), attained near \(\tau=0.204\));
let \(S'=X_1-X_2^2\), the first coordinate of the maximum likelihood estimator \(G^{-1}(X)\), which is also the one step summary from any preliminary estimate with \(\bar\theta_2=X_2\), and let \(S_G=Y_{G,1}/\sqrt N\) in the Gaussian model at \(\theta_\ast=0\), coupled as in (a); then \(\sqrt N\,E\,\mathscr{W}_1(\Pi(\cdot\mid S'),\Pi_G(\cdot\mid S_G))\to0\), so both hypotheses hold for \(S'\) with this model and coupling.
Source: source/chapters/ext/m2c.tex, label m:prop:gaptoy.
Effective dimension under dissipation
Theorem 15.46, page 267 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on Assumption U and other hypotheses of the quantile generator theorem, in part (ii) for nonlinear maps.
Statement. Consider the heat equation, or the Stokes equation on divergence free fields, on the torus \(\mathbb{T}^d=[0,1]^d\) with viscosity \(\nu\), Laplace eigenpairs \((e_j,\lambda_j)\) with \(\lambda_j=4\pi^2|k|^2\) for the \(j\)th lattice mode \(k\in\mathbb{Z}^d\setminus\{0\}\), \(N\) noisy observations of the solution at points uniform on \([0,T]\times\mathbb{T}^d\), per mode information \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\), and prior variances \(\sigma^2_{0,j}=\rho^2(\lambda_j/\lambda_1)^{-\alpha}\) for the coefficients \(\langle e_j,\theta\rangle\) of the initial condition. Define the effective dimension \(J_{\rm eff}=\#\{j:N\iota_j\sigma^2_{0,j}\ge1\}\), the number of modes in which the data at least halve the prior variance. Then:
\(J_{\rm eff}=\#\{j:\lambda_j^{\alpha+1}\le c_jN\rho^2\lambda_1^\alpha/(2\nu T)\}\) with \(c_j=1-e^{-2\nu\lambda_jT}\in[1-e^{-2\nu\lambda_1T},1]\); hence, as \(\Lambda=(N\rho^2\lambda_1^\alpha/(2\nu T))^{1/(\alpha+1)}\to\infty\) with \(\nu T\Lambda\ge1\), and with constants that depend only on \(\alpha\), \(d\) and \(\rho\), \(J_{\rm eff}\asymp(N\rho^2/(\nu T))^{d/(2(\alpha+1))}\asymp N^{d/(2\alpha+2)}(\nu T)^{-d/(2\alpha+2)}\). For \(d=2\) this is \(N^{1/(\alpha+1)}\), which for \(\alpha=\gamma\) is the order of the dimension \(J_N=N\delta_N^2=N^{1/(\gamma+1)}\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\), of the summary in Nickl’s (2024) Bernstein von Mises theorem; at fixed \(N\), viscosity or observation time reduces the effective dimension polynomially once \(\nu T\ge\lambda_1^{-1}(2/(N\rho^2))^{1/\alpha}\), and for every \(\nu T\) the count is at most \(\#\{j:\lambda_j\le\lambda_1(N\rho^2)^{1/\alpha}\}\), because \(\iota_j\le1\).
In the linear model (the model truncated to \(D\) modes, \(Y=\Phi\theta+\varepsilon\) with unit noise variance, prior covariance \(\Sigma_0\) and posterior covariance \(V_N=(\Phi^\top\Phi+\Sigma_0^{-1})^{-1}\)), the identity skip generator, which adds to the summary a fixed linear map of standard Gaussian noise, is exact to leading order with the analytic scale \((N\iota_j+\sigma_{0,j}^{-2})^{-1/2}\), is exact with the scale \(V_N^{1/2}\), and requires no training. For a nonlinear forward map whose information operator has the form \(\mathbb{I}^*\mathbb{I}=\mathcal{L}^*\mathcal{L}+\mathcal{K}\), with \(\mathcal{L}\) the Stokes (heat) solution operator and \(\mathcal{K}\) compact and gaining two derivatives, assume the hypotheses of the book’s Bernstein von Mises theorem for the identity skip quantile generator, namely Conditions 1 and 2 of Nickl (2024), the uniform Bernstein von Mises assumption for the one step summary \(\hat S_N\), the whitened training residual assumption (U2’), the hypotheses under which \(\hat S_N\) is \(\sqrt N\) equivalent to the efficient centering, and error bounds on \(\hat S_N\). Write the limit covariance on the first \(J_N\) modes as \(\Sigma=\mathcal{D}^{1/2}(\mathrm{Id}+\Xi)\mathcal{D}^{1/2}\) with \(\mathcal{D}=\operatorname{diag}(1/\iota_j)\), where \(\Xi\) is the compression to the first \(J_N\) modes of the compact operator \(\Xi_\infty=(\mathrm{Id}+K')^{-1}-\mathrm{Id}\), \(K'=T^{-1}\mathcal{D}^{1/2}\mathcal{K}\mathcal{D}^{1/2}\), so that \(\|\Xi P_{>J}\|_{op}\le\|\Xi_\infty P_{>J}\|_{op}\to0\) as \(J\to\infty\) uniformly in \(N\), with \(P_{>J}\) the projection onto the modes \(j>J\). The generator of that theorem is \(\widehat G(s,U)=s+N^{-1/2}\mathcal{D}^{1/2}\hat C_M^{1/2}q(U)\), with \(U\) uniform on \([0,1]^{J_N}\), \(q\) the standard normal quantile function applied coordinatewise, and \(\hat C_M\) the uncentered second moment matrix of the whitened training residuals \(\mathcal{D}^{-1/2}\sqrt N(\theta^{(m)}-\hat S^{(m)}_N)\), \(m\le M\). Given a cut \(J'=J'_N\le J_N\) with \(J'\to\infty\), the hybrid generator replaces the head \(\hat C_M^{1/2}\) by \((\hat C^{\rm hyb}_M)^{1/2}\), where \(\hat C^{\rm hyb}_M=\operatorname{diag}(\hat C^{J'}_M,\mathrm{Id})\) and \(\hat C^{J'}_M\) is the second moment matrix of the first \(J'\) whitened coordinates, so that it learns the correction to the analytic scale \(\mathcal{D}^{1/2}\) on the first \(J'\) modes only and uses the analytic scale beyond. Then \(\|\hat C^{\rm hyb}_M-(\mathrm{Id}+\Xi)\|_{op}\le O_P\big(\sqrt{J'/M}+J'/M\big)+2\|\Xi_\infty P_{>J'}\|_{op}+o_P(1)\), and the conclusion of that theorem (the GBC posterior \(\hat\Pi\) satisfies \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}(\hat\Pi,\Pi_N)\to0\) in \(P^N_{\theta_0}\) probability, \(\Pi_N\) being the posterior and \(\mathscr{W}_{1,H^{-k-2}}\) the Wasserstein distance of order one on the negative Sobolev space \(H^{-k-2}\), together with the tail and moment condition (G2) of the book’s transfer theorem) holds for the hybrid generator with \(J'/M_N\to0\) in place of \(J_N/M_N\to0\); with the cut \(J'=\min(J_{\rm eff},J_N)\) it suffices that \(J_{\rm eff}/M_N\to0\).
In the Gaussian experiment that observes \(\theta_j+(N\iota_j)^{-1/2}\xi_j\) with independent standard normal \(\xi_j\) and independent priors \(\theta_j\sim N(0,\sigma^2_{0,j})\), a generator that reproduces the posterior center exactly but uses the prior spread for the modes \(j>J'\) incurs a Wasserstein penalty, of order one in the Euclidean norm, of at most \(\sqrt{2/\pi}\sum_{j>J'}\sigma_{0,j}\big(1-(1+N\iota_j\sigma^2_{0,j})^{-1/2}\big)\le\sqrt{1/(2\pi)}\sum_{j>J'}N\iota_j\sigma^3_{0,j}\), which decays like \((J')^{1-(3\alpha+2)/d}\) when the sum is replaced by its integral.
Source: source/chapters/ext/m2c.tex, label m:thm:jeff.
Sufficiency versus the raw data: simulation cost of a generator trained without a summary
Theorem 15.47, page 269 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on Assumption U and other hypotheses of the quantile generator theorem, for the second claim only.
Statement. Let a generator be trained on \(M\) i.i.d. simulated pairs to reproduce the conditional law of \(\theta\) given the raw observation vector \(Y\in\mathbb{R}^N\) (or \(\mathbb{R}^{2N}\) for a vector field), with no summary, and let it be required to be accurate uniformly over the class \(\mathcal{Q}_\beta(N)\) of conditional quantile maps \((\tau,s)\mapsto Q(\tau\mid s)\) on \([0,1]\times[0,1]^N\) that have Hölder norm of order \(\beta\) at most one and are nondecreasing in \(\tau\), the inputs having a law on \([0,1]^N\) with a Lebesgue density bounded above by \(C_\mu\) and below by a positive constant. Then the number of simulations needed for \(\sqrt N\,E\,\mathscr{W}_1(\hat\Pi,\Pi(\cdot\mid Y))\to0\), with \(\hat\Pi\) the law the generator reports and \(\mathscr{W}_1\) the Wasserstein distance of order one, satisfies \(M\ge cN^{(2\beta+N)/(2\beta)}\) for all large \(N\), where \(c=C_\mu^{-1}\); the right side grows faster than any exponential in \(N\). For the heat or Stokes model with prior variances \(\sigma^2_{0,j}=\rho^2(\lambda_j/\lambda_1)^{-\alpha}\), viscosity \(\nu\), observation window \([0,T]\) and effective dimension \(J_{\rm eff}=\#\{j:N\iota_j\sigma^2_{0,j}\ge1\}\asymp N^{d/(2\alpha+2)}(\nu T)^{-d/(2\alpha+2)}\) (catalog entry effective_dimension_under_dissipation), the hybrid generator of part (ii) of that entry with the cut \(\min(J_{\rm eff},J_N)\), trained on the sufficient statistic, which uses the Bernstein von Mises structure that the uniform class does not, satisfies the accuracy condition (G1), \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}(\hat\Pi,\Pi_N)\to0\) in \(P^N_{\theta_0}\) probability with \(\Pi_N\) the posterior, as soon as \(J_{\rm eff}/M\to0\), conditionally on the hypotheses of part (ii) of that entry, which include those of the book’s Theorem 16.10 and the uniform Bernstein von Mises assumption.
Source: source/chapters/ext/m2c.tex, label m:thm:raw.
Chapter 16. Transfer, Generators and Localized Training
Transfer of Bernstein von Mises guarantees to the GBC posterior (BvM for GBC)
Theorem 16.2, page 275 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Setting. Let \(\Omega=[0,1]^d\) be the torus, \(T>0\) and \(\mathcal{X}=[0,T]\times\Omega\). The data are \(Z^{(N)}=(Y_i,X_i)_{i\le N}\) with \(Y_i=u_\theta(X_i)+\varepsilon_i\), where \(u_\theta=\mathscr{G}(\theta)\) solves an evolution equation with initial condition \(\theta\), the \(X_i\) are i.i.d. uniform on \(\mathcal{X}\) and the \(\varepsilon_i\sim N(0,I_W)\) are independent of them. Let \((e_j)\) be the Laplace eigenbasis, with eigenvalues \(\lambda_j\asymp j^{2/d}\); let the prior \(\Pi\) be the law of \((\sqrt N\delta_N)^{-1}\sum_j\lambda_j^{-\gamma/2}g_je_j\) with \(g_j\) i.i.d. \(N(0,1)\) and \(\delta_N=N^{-\gamma/(2\gamma+d)}\); let \(\Pi_N=\Pi(\cdot\mid Z^{(N)})\) be the posterior, \(\theta_0\) the true initial condition and \(P^N_{\theta_0}\) the law of the data. Let \(\mathbb{I}_{\theta_0}\) be the linearization of \(\mathscr{G}\) at \(\theta_0\) and \(\mathcal{N}_{\theta_0}=N(0,T(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1})\). Assume Conditions 1 and 2 of Nickl (2024): the prior above and, for the forward map on Sobolev balls, (A) uniform \(L^\infty\) bounds, (B) \(L^2\) Lipschitz continuity, (C) a linearization \(\mathbb{I}_{\theta_0}\) with quadratic remainder, (D) \(L^\infty\) mapping properties from \(H^\zeta\) with \(\zeta>d/2\), (E) the stability estimate \(\|\mathscr{G}(\theta)-\mathscr{G}(\theta_0)\|_{L^2(\mathcal{X})}\ge c\|\theta-\theta_0\|_{H^{-1}}\) and (F) invertibility of the information operator (\(\Delta\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) a homeomorphism of \(H^\eta_0\) for \(\eta\ge\eta_0\)), with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\).
Accuracy conditions. Let \(\mathscr{W}_1\) be the Wasserstein distance of order one on \(H^{-k-2}(\Omega)\). Let \(\bar\gamma\) be the Sobolev index of the localization ball, chosen as in the proofs of Nickl (2024) and of Konen and Nickl (2026) in \(\max(2,2\zeta)<\bar\gamma<\gamma-\max(1,d/2)\); put \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\), \(\bar\Theta_{N,L}=\{\|\theta\|_{H^{\bar\gamma}}\le L\}\cap\{\|\theta-\theta_0\|_{H^\xi}\le L\tilde\delta_N(\xi)\ \text{for all}\ 0\le\xi<\bar\gamma\}\), and \(q=4\) in the setting of Theorem 1 of Nickl (2024), \(q=4\bar\gamma\) in that of Theorem 1.1 of Konen and Nickl (2026). Let \(\widehat\Pi_N\) be the GBC posterior (the law of the generator’s output at the observed data, a measurable function of \(Z^{(N)}\) and of simulation randomness drawn independently of \(Z^{(N)}\)) and \(\hat\theta^{G}_N\) its mean. Assume that, in \(P^N_{\theta_0}\) probability as \(N\to\infty\): (G1) \(\sqrt N\,\mathscr{W}_1(\widehat\Pi_N,\Pi_N)\to0\); (G2) for some \(b>0\) and \(L\), \(\widehat\Pi_N(\bar\Theta_{N,L}^c)\le e^{-bN\delta_N^2}\) and \(\mathbb{E}^{\widehat\Pi_N}\|\theta\|^q_{H^{\bar\gamma}}=O_P(1)\).
Then: (i) \(\mathscr{W}_1\big(\operatorname{Law}(\sqrt N(\theta-\hat\theta^{G}_N)\mid\theta\sim\widehat\Pi_N),\mathcal{N}_{\theta_0}\big)\to0\) in \(P^N_{\theta_0}\) probability, and \(\sqrt N(\hat\theta^{G}_N-\theta_0)\to\mathcal{N}_{\theta_0}\) in distribution in \(H^{-k-2}\). (ii) If in addition the forward map satisfies the hypotheses of Theorem 1 of Nickl (2024) (the reaction diffusion equation \(\partial_tu-\Delta u=f(u)\) with \(f\in C^\infty_c(\mathbb{R})\), \(d\le3\), integer \(\gamma>2+3d\) and \(\theta_0\) smooth, with \(\mathscr{C}=C([t_{\min},t_{\max}],C(\Omega))\) for \(0<t_{\min}<t_{\max}<\infty\)) or of Theorem 1.1 of Konen and Nickl (2026) (the two dimensional Navier Stokes equations, with the space of trajectories \(\mathscr{C}\) of that theorem), and \(\mu\) is the law in \(\mathscr{C}\) of the solution \(U\) of the equation linearized at \(u_{\theta_0}\) with \(U(0)\sim\mathcal{N}_{\theta_0}\), then \(\mathscr{W}_{1,\mathscr{C}}\big(\operatorname{Law}(\sqrt N(u_\theta-u_{\hat\theta^{G}_N})\mid\theta\sim\widehat\Pi_N),\mu\big)\to0\) in probability and \(\sqrt N(u_{\hat\theta^{G}_N}-u_{\theta_0})\to\mu\) in distribution in \(\mathscr{C}\). (iii) Under (ii), for \(\Phi:\mathscr{C}\to\mathbb{B}\) Hadamard differentiable into a Banach space \(\mathbb{B}\), \(\sqrt N(\Phi(u_{\hat\theta^{G}_N})-\Phi(u_{\theta_0}))\to\dot\Phi_{u_{\theta_0}}[U]\) in distribution, \(U\sim\mu\), and the GBC credible band \(C^{G}_N=\{u\in\mathscr{C}:\|u-u_{\hat\theta^{G}_N}\|_{\mathscr{C}}\le R_N\}\), with \(R_N\) the \(1-\beta\) quantile of \(\|u_\theta-u_{\hat\theta^{G}_N}\|_{\mathscr{C}}\) under \(\theta\sim\widehat\Pi_N\), satisfies \(P^N_{\theta_0}(u_{\theta_0}\in C^{G}_N)\to1-\beta\), and \(\sqrt N R_N\to c>0\). (iv) In the Navier Stokes case of (ii), let \(\theta_{N,h}=\theta_0+h/\sqrt N\), and let \(\dot H^{\infty}_\diamond\) be the smooth, divergence free, mean zero vector fields on the torus. For every \(h\in\dot H^{\infty}_\diamond\), under \(P^N_{\theta_{N,h}}\), \(\sqrt N(\hat\theta^{G}_N-\theta_{N,h})\to\mathcal{N}_{\theta_0}\) in distribution in \(H^{-k-2}\) and \(\sqrt N(u_{\hat\theta^{G}_N}-u_{\theta_{N,h}})\to\mu\) in distribution in \(\mathscr{C}\); that is, \(u_{\hat\theta^{G}_N}\) is a regular estimator sequence of the state at \(\theta_0\) whose limit law \(\mu\) attains the bound of Theorem 1.2 of Konen and Nickl (2026) (Theorem 2.10 there for a general \(\Phi\)). Consequently, with \(\Phi\) as in (iii) and \(\Phi=\operatorname{Id}\) permitted, \(\Phi(u_{\hat\theta^{G}_N})\) is efficient among regular estimator sequences by the convolution theorem, and for every bounded, continuous, subconvex loss \(\ell:\mathbb{B}\to[0,\infty)\) it attains the local asymptotic minimax bound: \[\sup_{J\subset\dot H^{\infty}_\diamond,\ |J|<\infty}\ \liminf_{N\to\infty}\ \max_{h\in J}\ \mathbb{E}^N_{\theta_{N,h}}\,\ell\Big(\sqrt N\big(\Phi(u_{\hat\theta^{G}_N})-\Phi(u_{\theta_{N,h}})\big)\Big)=\mathbb{E}\,\ell\big(\dot\Phi_{u_{\theta_0}}[U]\big),\qquad U\sim\mu.\]
Source: source/chapters/ext/m3a.tex, label m:thm:main.
Solve then simulate equals simulate the limit: tangent linear forward simulation of GBC draws
Corollary 16.4, page 280 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on the uniform Bernstein von Mises assumption (Assumption U), for the plug in bound only.
Statement. Let \(\widehat\Pi_N\) be a GBC posterior with mean \(\hat\theta^{G}_N\) that satisfies the hypotheses of part (ii) of the transfer theorem (catalog entry bvm_for_gbc): data \(Y_i=u_\theta(X_i)+\varepsilon_i\) on \(\mathcal{X}=[0,T]\times\Omega\) with Nickl’s rescaled Gaussian series prior and posterior \(\Pi_N\), Conditions 1 and 2 of Nickl (2024) with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\), the accuracy conditions (G1) \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}(\widehat\Pi_N,\Pi_N)\to0\) and (G2) (mass at most \(e^{-bN\delta_N^2}\) off the localization set and \(\mathbb{E}^{\widehat\Pi_N}\|\theta\|^q_{H^{\bar\gamma}}=O_P(1)\), with \(\bar\gamma\), \(\delta_N\) and \(q\) as in that entry), and the hypotheses of Theorem 1 of Nickl (2024) (reaction diffusion) or of Theorem 1.1 of Konen and Nickl (2026) (two dimensional Navier Stokes), with \(\mathscr{C}\) the space of trajectories of those theorems. Let \(\mathbb{I}_\vartheta\) be the linearization of \(\theta\mapsto u_\theta\) at \(\vartheta\), \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\), and let \(\mu\) be the law of the solution of the tangent linear equation linearized at \(u_{\theta_0}\) with initial datum drawn from \(\mathcal{N}_{\theta_0}\). Then the law of \(\sqrt N(u_\theta-u_{\hat\theta^{G}_N})\) obtained by pushing \(\theta\sim\widehat\Pi_N\) through the nonlinear solver and the law \(\mu\) obtained by pushing \(\mathcal{N}_{\theta_0}\) through the tangent linear equation are asymptotically equal in \(\mathscr{W}_{1,\mathscr{C}}\).
Suppose in addition that Assumption U (uniform Bernstein von Mises) holds with a set \(\Theta_N\) that contains \(\hat\theta^{G}_N\) with probability tending to one: for the summary \(S_N\), (U1) \(\Pi(\Theta_N)\to1\), \(\theta_0\in\Theta_N\) for all large \(N\) and \(\sup_{\vartheta\in\Theta_N}\mathbb{E}^N_\vartheta\,\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_\vartheta\big)\to0\); (U2) these discrepancies are uniformly integrable under the prior predictive law; (U3) \(\vartheta\mapsto T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}\) is Lipschitz from \((\Theta_N,\|\cdot\|_{H^\zeta})\) into bounded operators \(H^{-k-2}\to H^{-k-4}\) and \(\vartheta\mapsto\mathcal{N}_\vartheta\) is Lipschitz from \((\Theta_N,\|\cdot\|_{H^\zeta})\) into the laws on \(H^{-k-2}\) with the distance \(\mathscr{W}_{1,H^{-k-2}}\), with constants free of \(N\), the first moments \(m_1(\vartheta)=\int\|x\|_{H^{-k-2}}\,\mathrm{d}\mathcal{N}_\vartheta(x)\) satisfy \(\int_{\Theta_N^c}m_1\,\mathrm{d}\Pi\to0\) and \(\Pi(\Theta_N^c)\sup_{\Theta_N}m_1\to0\), \(S_N\in\Theta_N\) with probability tending to one and \(\mathbb{E}^\Pi[\|\theta-S_N\|_{H^\zeta}\mid S_N]\to0\); (U4) for every \(\theta_0\in\Theta_N\), \(\mathbb{E}\big[\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\mid S_N\big]\to0\) in \(P^N_{\theta_0}\) probability. Suppose also that, for the index \(\zeta>d/2\) of Condition 2D of Nickl (2024), \(\vartheta\mapsto\mathbb{I}_\vartheta\) is Lipschitz from an \(H^\zeta\) ball about \(\theta_0\) into the bounded operators \(H^{-k-2}\to\mathscr{C}\). Then \[\varrho_N:=\|\hat\theta^{G}_N-\theta_0\|_{H^\zeta}=O_P\big(N^{-(\bar\gamma-\zeta)/(2(\bar\gamma+k+2))}\big)\to0,\] and, writing \(U^\vartheta\) for the solution of the tangent equation linearized at \(u_\vartheta\) with initial datum drawn from \(\mathcal{N}_\vartheta\) (so that \(U^{\theta_0}\) has law \(\mu\)), the forward simulation step may replace the nonlinear solves by the tangent equation linearized at \(\hat\theta^{G}_N\) with initial law \(N(0,T(\mathbb{I}^*_{\hat\theta^{G}_N}\mathbb{I}_{\hat\theta^{G}_N})^{-1}/N)\), with plug in error \[\mathscr{W}_{1,\mathscr{C}}\Big(\operatorname{Law}\big(\sqrt N(u_\theta-u_{\hat\theta^{G}_N})\mid\theta\sim\widehat\Pi_N\big),\ \operatorname{Law}\big(U^{\hat\theta^{G}_N}\big)\Big)\le o_P(1)+C\varrho_N,\] where \(C\) depends only on the Lipschitz constant of \(\vartheta\mapsto\mathbb{I}_\vartheta\), on the Lipschitz constant of \(\vartheta\mapsto\mathcal{N}_\vartheta\) in \(\mathscr{W}_{1,H^{-k-2}}\) from (U3), on the norm of \(\mathbb{I}_{\theta_0}:H^{-k-2}\to\mathscr{C}\) and on the first moment of \(\mathcal{N}_{\theta_0}\) in \(H^{-k-2}\).
Source: source/chapters/ext/m3a.tex, label m:cor:commute.
Bernstein von Mises for the identity skip Gaussian head quantile generator, conditional on the uniform hypothesis
Theorem 16.10, page 285 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: conditional on Assumption U, the whitened residual assumption (U2’) and error bounds on the summary.
Statement. Setting. Data \(Y_i=u_\theta(X_i)+\varepsilon_i\), \(i\le N\), with \(X_i\) i.i.d. uniform on \(\mathcal{X}=[0,T]\times\Omega\), \(\Omega=[0,1]^d\) the torus, \(\varepsilon_i\sim N(0,I_W)\) independent, \(u_\theta\) the solution of an evolution equation with initial condition \(\theta\); \((e_j)\) the Laplace eigenbasis with eigenvalues \(\lambda_j\asymp j^{2/d}\); the prior \(\Pi\) the law of \((\sqrt N\delta_N)^{-1}\sum_j\lambda_j^{-\gamma/2}g_je_j\), \(g_j\) i.i.d. \(N(0,1)\), \(\delta_N=N^{-\gamma/(2\gamma+d)}\); \(\Pi_N\) the posterior, \(\theta_0\) the truth, \(P^N_{\theta_0}\) the law of the data \(Z^{(N)}\); \(\mathbb{I}_\vartheta\) the linearization of \(\theta\mapsto u_\theta\) at \(\vartheta\) and \(\mathcal{N}_\vartheta=N(0,T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1})\). Assume Conditions 1 and 2 of Nickl (2024) (as in catalog entry bvm_for_gbc) with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\), \(\zeta>d/2\) the index of Condition 2D; let \(\bar\gamma\) satisfy \(\max(2,2\zeta)<\bar\gamma<\gamma-\max(1,d/2)\) and \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\).
Summary and generator. Let \(J=J_N\asymp N\delta_N^2\), \(p_N\) the projection onto the first \(J\) modes, and \(\hat S_N\in\mathbb{R}^J\) the cross fitted one step efficient score summary: the sample is split into halves \(\mathcal{I}_1,\mathcal{I}_2\), preliminary estimates \(\bar\theta^{[b]}_N\) computed from half \(b\) satisfy \(\|\bar\theta^{[b]}_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for \(0\le\xi\le\bar\gamma\), and \(\hat S_{N,j}=\frac12\sum_{b=1,2}\big[\langle e_j,\bar\theta^{[b]}_N\rangle+|\mathcal{I}_{3-b}|^{-1}\sum_{i\in\mathcal{I}_{3-b}}\langle\mathbb{I}_{\bar\theta^{[b]}_N}p_N(T(\mathbb{I}^*_{\bar\theta^{[b]}_N}\mathbb{I}_{\bar\theta^{[b]}_N})^{-1}e_j)(X_i),Y_i-u_{\bar\theta^{[b]}_N}(X_i)\rangle_W\big]\) for \(j\le J\). Let \(\mathcal{D}=\operatorname{diag}(1/\iota_j)=T(\mathcal{L}^*\mathcal{L})^{-1}\) be the information diagonal of the heat (or Stokes) solution operator \(\mathcal{L}\), \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\), and write \(\Sigma=T(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1}\) restricted to the first \(J\) modes as \(\Sigma=\mathcal{D}^{1/2}(\operatorname{Id}+\Xi)\mathcal{D}^{1/2}\). From \(M\) simulated pairs \((\theta^{(m)},Z^{(m)})\) with summaries \(\hat S^{(m)}_N\) form the whitened residuals \(W^{(m)}=\mathcal{D}^{-1/2}\sqrt N(\theta^{(m)}-\hat S^{(m)}_N)\in\mathbb{R}^J\) and \(\hat C_M=M^{-1}\sum_mW^{(m)}W^{(m)\top}\), and let \(\widehat G(s,U)=s+N^{-1/2}\mathcal{D}^{1/2}\hat C_M^{1/2}\Phi^{-1}(U)\), \(U\sim\mathrm{Unif}[0,1]^J\), with \(\Phi^{-1}\) the standard normal quantile function applied coordinatewise; the GBC posterior \(\widehat\Pi_N\) is the law of \(\widehat G(\hat S_N,U)\) given the data and the training sample, that is \(N(\hat S_N,N^{-1}\mathcal{D}^{1/2}\hat C_M\mathcal{D}^{1/2})\).
Hypotheses. (a) Assumption U with \(S_N=\hat S_N\), clause (U4) included: there is a set \(\Theta_N\) with (U1) \(\Pi(\Theta_N)\to1\), \(\theta_0\in\Theta_N\) for all large \(N\) and \(\sup_{\vartheta\in\Theta_N}\mathbb{E}^N_\vartheta\,\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_\vartheta\big)\to0\); (U2) these discrepancies are uniformly integrable under the prior predictive law; (U3) \(\vartheta\mapsto T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}\) is Lipschitz from \((\Theta_N,\|\cdot\|_{H^\zeta})\) into bounded operators \(H^{-k-2}\to H^{-k-4}\) and \(\vartheta\mapsto\mathcal{N}_\vartheta\) is Lipschitz from \((\Theta_N,\|\cdot\|_{H^\zeta})\) into the laws on \(H^{-k-2}\) with \(\mathscr{W}_{1,H^{-k-2}}\), with constants free of \(N\), the first moments \(m_1(\vartheta)=\int\|x\|_{H^{-k-2}}\,\mathrm{d}\mathcal{N}_\vartheta(x)\) satisfy \(\int_{\Theta_N^c}m_1\,\mathrm{d}\Pi\to0\) and \(\Pi(\Theta_N^c)\sup_{\Theta_N}m_1\to0\), \(S_N\in\Theta_N\) with probability tending to one and \(\mathbb{E}^\Pi[\|\theta-S_N\|_{H^\zeta}\mid S_N]\to0\); (U4) for every \(\theta_0\in\Theta_N\), \(\mathbb{E}\big[\mathscr{W}_{1,H^{-k-2}}\big(\operatorname{Law}(\sqrt N(\theta-S_N)\mid Z^{(N)}),\mathcal{N}_{\theta_0}\big)\mid S_N\big]\to0\) in \(P^N_{\theta_0}\) probability. (b) Assumption (U2’): \(W^{(m)}=W^{(m)}_{g}+W^{(m)}_{r}\) with the pairs \((W^{(m)}_{g},W^{(m)}_{r})\) i.i.d. given \(Z^{(N)}\), \(\varkappa_N=\sup_{\|v\|_2=1}\|\langle v,W^{(m)}_{g}\rangle\|_{\psi_2\mid Z^{(N)}}=O_P(1)\) under \(P^N_{\theta_0}\), where \(\|\chi\|_{\psi_2\mid Z^{(N)}}=\inf\{K>0:\mathbb{E}[e^{\chi^2/K^2}\mid Z^{(N)}]\le2\}\), and, in \(P^N_{\theta_0}\) probability, \(\mathbb{E}[\|W^{(m)}_{r}\|_2^2\mid Z^{(N)}]\to0\) and \(\|\mathbb{E}[W^{(m)}_{g}W^{(m)\top}_{g}\mid Z^{(N)}]-(\operatorname{Id}+\Xi)\|_{op}\to0\). (c) The hypotheses of the equivalence of the one step summary with the efficient centering (catalog entry equivalence_with_the_efficient_centering), its two conditions on the exponents included: \(\sqrt N\,\tilde\delta_N(0)\tilde\delta_N(\zeta)\to0\), \(\gamma<(\bar\gamma+1)(\kappa-d/2)\) for some \(2d+\zeta<\kappa<k-d/2\) unless \(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0}\) is diagonal in \((e_j)\) or the preliminary estimates converge at the rate \(\delta_N\) in \(H^{-1}\), and, for every \(B>0\), a constant \(L(B)\) with \(\|\mathbb{I}_\vartheta p_N(T(\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta)^{-1}e_j)-\mathbb{I}_{\theta_0}p_N(T(\mathbb{I}^*_{\theta_0}\mathbb{I}_{\theta_0})^{-1}e_j)\|_{L^2(\mathcal{X})}\le L(B)\|\vartheta-\theta_0\|_{H^\zeta}\|e_j\|_{H^{\kappa+2}}\) for all \(\vartheta\) with \(\|\vartheta\|_{H^{\bar\gamma}}\le B\) and all \(j\le J_N\). (d) \(\|\hat S_N-\theta_0\|_{L^2}=o_P(\tilde\delta_N(0))\) and \(\|\hat S_N-\theta_0\|_{H^{\bar\gamma}}=o_P(1)\).
Conclusion. If \(M=M_N\) satisfies \(J_N/M_N\to0\), then \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}(\widehat\Pi_N,\Pi_N)\to0\) in \(P^N_{\theta_0}\) probability (condition (G1) of the transfer theorem) and condition (G2) of that theorem holds (mass at most \(e^{-bN\delta_N^2}\) off the localization set and \(\mathbb{E}^{\widehat\Pi_N}\|\theta\|^q_{H^{\bar\gamma}}=O_P(1)\)). Consequently the conclusions of the transfer theorem (catalog entry bvm_for_gbc) hold for the GBC posterior produced by this generator, those of its parts (ii) to (iv) under the additional hypotheses stated there.
Source: source/chapters/ext/m3b.tex, label m:thm:qnnA.
No uniform guarantee: minimax lower bound for conditional quantile generators over monotone Hölder classes
Theorem 16.15, page 290 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>0\) and \(J\in\mathbb{N}\) let \(\mathcal{Q}_\beta(J)\) be the set of maps \((\tau,s)\mapsto Q(\tau\mid s)\) from \([0,1]\times[0,1]^J\) to \(\mathbb{R}\) such that (i) \(\|Q\|_{C^\beta}\le1\), where, with \(\ell\) the largest integer strictly below \(\beta\), \(\|Q\|_{C^\beta}=\max_{|\alpha|\le\ell}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=\ell}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-\ell}\), the maxima running over multi indices on \(\mathbb{R}^{1+J}\); (ii) \(\tau\mapsto Q(\tau\mid s)\) is nondecreasing for every \(s\). Let \(P_{Q,s}=Q(\cdot\mid s)_\#\mathrm{Unif}[0,1]\) be the law that \(Q\) encodes at \(s\) and \(\mathscr{W}_{1,\mathbb{R}}\) the Wasserstein distance of order one on the line. The training sample consists of \(M\) i.i.d. copies of \((S,\theta)\) with \(S\sim\mu\) and \(\theta\mid S\sim P_{Q,S}\), where the design law \(\mu\) on \([0,1]^J\) has a Lebesgue density bounded above by \(C_\mu\) and below by \(c_\mu>0\) (so \(C_\mu\ge1\)). A generator is any measurable, possibly randomized, map \(\hat P_M\) from the training sample and a summary value \(s\) to a probability measure \(\hat P_M(\cdot\mid s)\) on \(\mathbb{R}\).
Let \(\beta>0\), \(J\ge1\), and let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary. There is a constant \(c=c(\beta)>0\), depending only on \(\beta\) and on two fixed profile functions, and in particular not on \(J\), such that for every \(M\ge1\) and every generator \(\hat P_M\), \[\sup_{Q\in\mathcal{Q}_\beta(J)}P\Big(\mathscr{W}_{1,\mathbb{R}}\big(\hat P_M(\cdot\mid s_0),P_{Q,s_0}\big)\ge c\,(C_\mu M)^{-\beta/(2\beta+J)}\Big)\ge\frac18,\] the probability being over the \(M\) training pairs, and consequently \[\inf_{\hat P_M}\ \sup_{Q\in\mathcal{Q}_\beta(J)}\mathbb{E}\,\mathscr{W}_{1,\mathbb{R}}\big(\hat P_M(\cdot\mid s_0),P_{Q,s_0}\big)\ge\frac c8\,(C_\mu M)^{-\beta/(2\beta+J)}.\] Consequently, if a GBC generator trained on \(M_N\) pairs is required to satisfy the accuracy condition (G1) of the transfer theorem at the summary value \(s_0\) uniformly over \(\mathcal{Q}_\beta(J_N)\), that is, if \(\sqrt N\,\mathscr{W}_{1,\mathbb{R}}(\hat P_{M_N}(\cdot\mid s_0),P_{Q,s_0})\to0\) in probability is to hold for every conditional quantile map in the class (reading this as (G1), which compares with the posterior given all the data, uses the asymptotic sufficiency of the summary and so Assumption U of the book), then necessarily \(\sqrt N\,(C_\mu M_N)^{-\beta/(2\beta+J_N)}\to0\), hence \[M_N\ge C_\mu^{-1}N^{(2\beta+J_N)/(2\beta)}=C_\mu^{-1}N^{1+J_N/(2\beta)}\qquad\text{for all large }N,\] which grows faster than any polynomial in \(N\) as soon as \(J_N\to\infty\).
Source: source/chapters/ext/m3b.tex, label m:thm:qnnB.
The minimax exponent for conditional quantile generators is attained for Hölder smoothness at most one (histogram generator)
Proposition 16.16, page 293 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>0\) and \(J\in\mathbb{N}\) let \(\mathcal{Q}_\beta(J)\) be the set of maps \((\tau,s)\mapsto Q(\tau\mid s)\) from \([0,1]\times[0,1]^J\) to \(\mathbb{R}\) with \(C^\beta\) norm at most one (for \(\beta\le1\), \(\|Q\|_\infty+\sup_{z\ne z'}|Q(z)-Q(z')|/\|z-z'\|_2^\beta\le1\) on \([0,1]^{1+J}\)) that are nondecreasing in \(\tau\) for every \(s\); let \(P_{Q,s}=Q(\cdot\mid s)_\#\mathrm{Unif}[0,1]\), let \(\mathscr{W}_{1,\mathbb{R}}\) be the Wasserstein distance of order one on the line, and let the training pairs \((S^{(i)},\theta^{(i)})\), \(i\le M\), be i.i.d. with \(S\sim\mu\) and \(\theta\mid S\sim P_{Q,S}\), where \(\mu\) has a Lebesgue density on \([0,1]^J\) bounded below by \(c_\mu>0\) and above by \(C_\mu\).
Let \(\beta\in(0,1]\), partition \([0,1]^J\) into \(m^J\) cubes of side \(h=1/m\) with \(m=\max\{1,\lfloor M^{1/(2\beta+J)}\rfloor\}\), and let \(\hat P_M(\cdot\mid s)\) be the empirical law of \(\{\theta^{(i)}:S^{(i)}\in B(s)\}\), where \(B(s)\) is the cube containing \(s\), and any fixed law on \([-1,1]\) if that cube contains no \(S^{(i)}\). Then \[\sup_{Q\in\mathcal{Q}_\beta(J)}\ \sup_{s\in[0,1]^J}\ \mathbb{E}\,\mathscr{W}_{1,\mathbb{R}}\big(\hat P_M(\cdot\mid s),P_{Q,s}\big)\le C(\beta,c_\mu)\,J^{\beta/2}\,M^{-\beta/(2\beta+J)},\] with \(C(\beta,c_\mu)\) depending only on \(\beta\) and \(c_\mu\). In particular, at fixed \(\beta\) and \(J\) a single explicit estimator has risk \(O(M^{-\beta/(2\beta+J)})=o(M^{-\beta/(2\beta+J+1)})\), so no lower bound with the exponent \(\beta/(2\beta+J+1)\) and a positive constant can hold: it is the summary dimension, and not the summary dimension plus one, that governs the problem.
Source: source/chapters/ext/m3b.tex, label m:prop:qnnUB.
The minimax exponent for conditional quantile generators is attained for smoothness between one and two by jittered local linear quantile regression
Proposition 16.19, page 295 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>0\) and \(J\in\mathbb{N}\) let \(\mathcal{Q}_\beta(J)\) be the set of maps \((\tau,s)\mapsto Q(\tau\mid s)\) from \([0,1]\times[0,1]^J\) to \(\mathbb{R}\) with \(\|Q\|_{C^\beta}\le1\), where, with \(\ell\) the largest integer strictly below \(\beta\), \(\|Q\|_{C^\beta}=\max_{|\alpha|\le\ell}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=\ell}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-\ell}\) (maxima over multi indices on \(\mathbb{R}^{1+J}\)), and with \(\tau\mapsto Q(\tau\mid s)\) nondecreasing for every \(s\). Let \(P_{Q,s}=Q(\cdot\mid s)_\#\mathrm{Unif}[0,1]\), \(\mathscr{W}_{1,\mathbb{R}}\) the Wasserstein distance of order one on the line, and let the training pairs \((S^{(i)},\theta^{(i)})\), \(i\le M\), be i.i.d. with \(S\sim\mu\) and \(\theta\mid S\sim P_{Q,S}\), where \(\mu\) has a Lebesgue density on \([0,1]^J\) between \(c_\mu>0\) and \(C_\mu\). Let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary.
Let \(\beta\in(1,2]\) and \(J\ge1\). Put \(h=\min\{1/8,M^{-1/(2\beta+J)}\}\), \(I=\{i\le M:\|S^{(i)}-s_0\|_2\le h\}\), and for \(i\in I\) let \(X_i=(S^{(i)}-s_0)/h\) and \(\tilde\theta_i=\theta^{(i)}+h^\beta V_i\), where \(V_1,\dots,V_M\) are i.i.d. uniform on \([0,1]\) and independent of the training pairs. For \(\tau\in(0,1)\) let \(\rho_\tau(r)=r(\tau-\mathbf{1}\{r<0\})\), and let \((\hat a_\tau,\hat b_\tau)\) be the minimizer of least Euclidean norm of \((a,b)\mapsto\sum_{i\in I}\rho_\tau(\tilde\theta_i-a-b\cdot X_i)\) over \(\mathbb{R}\times\mathbb{R}^J\) when the vectors \((1,X_i)\), \(i\in I\), span \(\mathbb{R}^{1+J}\), and \((\hat a_\tau,\hat b_\tau)=0\) otherwise. Let \(\hat Q(\tau)=\max\{-1,\min\{1,\hat a_\tau\}\}\) and \(\hat P_M(\cdot\mid s_0)=\hat Q_\#\mathrm{Unif}[0,1]\). There is a constant \(C\) depending only on \(\beta\), \(J\), \(c_\mu\) and \(C_\mu\) such that for every \(M\ge1\) \[\sup_{Q\in\mathcal{Q}_\beta(J)}\mathbb{E}\,\mathscr{W}_{1,\mathbb{R}}\big(\hat P_M(\cdot\mid s_0),P_{Q,s_0}\big)\le C\,M^{-\beta/(2\beta+J)}.\] In particular, for every \(\beta\le2\), at fixed \(\beta\) and \(J\) no lower bound with the exponent \(\beta/(2\beta+J+1)\) and a positive constant can hold.
Source: source/chapters/ext/m3b.tex, label m:prop:qnnUB2.
A local linear quantile fit is nearly monotone in its error (deterministic lemma for monotone Hölder quantile maps)
Lemma 16.21, page 296 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\beta\in(1,2]\), \(J\ge1\), and let \(Q:[0,1]\times[0,1]^J\to\mathbb{R}\), \((\tau,s)\mapsto Q(\tau\mid s)\), satisfy \(\|Q\|_{C^\beta}=\max_{|\alpha|\le1}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=1}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-1}\le1\) (multi indices on \(\mathbb{R}^{1+J}\)) and be nondecreasing in \(\tau\) for every \(s\). Let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary, \(h\in(0,1/8]\), \(\bar B=\{x\in\mathbb{R}^J:\|x\|_2\le1\}\), \(C_G=2\max\{(2\sqrt J)^{1/\beta},2^{3+1/\beta}\}\), \(K=2^{\beta-1}C_G^\beta\) and \(\eta=(K+\sqrt J+1)h^\beta\). For \(x\in\bar B\) let \(F_x\) be the distribution function of \(P_{Q,s_0+hx}=Q(\cdot\mid s_0+hx)_\#\mathrm{Unif}[0,1]\) and \[\tilde F_x(y)=h^{-\beta}\int_{y-h^\beta}^{y}F_x(t)\,\mathrm{d}t,\] the distribution function of \(\theta+h^\beta V\) for \(\theta\sim P_{Q,s_0+hx}\) and an independent \(V\sim\mathrm{Unif}[0,1]\); for \(\tau\in(0,1)\) let \(T_\tau(x)=Q(\tau\mid s_0)+h\nabla_sQ(\tau\mid s_0)\cdot x\). Let \(\tau\in(0,1)\), let \(p:\mathbb{R}^J\to\mathbb{R}\) be affine, and put \(w=p-T_\tau\) and \(\delta(x)=\tilde F_x(p(x))-\tau\). Then for all \(x,x'\in\bar B\) and \(t\ge0\): (i) if \(\delta(x)>0\) then \(w(x)\ge-\eta\), and if \(\delta(x)<0\) then \(w(x)\le\eta\); (ii) if \(w(x')\le t\) and \(w(x)>3t+4\eta\) then \(\delta(x')\le\delta(x)\), and if \(w(x')\ge-t\) and \(w(x)<-3t-4\eta\) then \(\delta(x)\le\delta(x')\); (iii) if \(w(x)>\eta\) then \(\delta(x)\ge\min\{\frac23(w(x)-\eta),1-\tau\}\), and if \(w(x)<-\eta\) then \(-\delta(x)\ge\min\{\frac23(-w(x)-\eta),\tau\}\).
Source: source/chapters/ext/m3b.tex, label m:lem:qrep.
Moment conditions force a small local linear quantile fit
Lemma 16.22, page 298 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\beta\in(1,2]\), \(J\ge1\), and let \(Q:[0,1]\times[0,1]^J\to\mathbb{R}\), \((\tau,s)\mapsto Q(\tau\mid s)\), satisfy \(\|Q\|_{C^\beta}=\max_{|\alpha|\le1}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=1}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-1}\le1\) (multi indices on \(\mathbb{R}^{1+J}\)) and be nondecreasing in \(\tau\) for every \(s\). Let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary, \(h\in(0,1/8]\), \(\bar B=\{x\in\mathbb{R}^J:\|x\|_2\le1\}\), \(C_G=2\max\{(2\sqrt J)^{1/\beta},2^{3+1/\beta}\}\), \(K=2^{\beta-1}C_G^\beta\) and \(\eta=(K+\sqrt J+1)h^\beta\). For \(x\in\bar B\) let \(F_x\) be the distribution function of \(Q(\cdot\mid s_0+hx)_\#\mathrm{Unif}[0,1]\) and \(\tilde F_x(y)=h^{-\beta}\int_{y-h^\beta}^yF_x(t)\,\mathrm{d}t\). Let \(\tau\in(0,1)\), \(T_\tau(x)=Q(\tau\mid s_0)+h\nabla_sQ(\tau\mid s_0)\cdot x\), \(p:\mathbb{R}^J\to\mathbb{R}\) affine, \(w=p-T_\tau\) and \(\delta(x)=\tilde F_x(p(x))-\tau\).
Let \(g\) be a probability density on \(\bar B\) with \(g\le g_+\), let \(\omega_{J-1}\) be the volume of the unit ball of \(\mathbb{R}^{J-1}\) (with \(\omega_0=1\)), and put \(C_R=1+\frac{20}{3}g_+\omega_{J-1}\) and \(c_B=12C_R(1+\sqrt J)\). Let \(\kappa\ge0\) be such that \(|\int\delta g|\le\kappa\) and \(|\int\delta(x)x_kg(x)\,\mathrm{d}x|\le\kappa\) for \(k=1,\dots,J\). If \(\min\{\tau,1-\tau\}>5c_B\kappa\), then \[\sup_{x\in\bar B}|w(x)|\le32\,C_R\,\eta+20\,C_R\,c_B\,\kappa.\]
Source: source/chapters/ext/m3b.tex, label m:lem:qbary.
The minimax exponent for conditional quantile generators is attained for smoothness above two when the conditional densities are bounded
Proposition 16.25, page 302 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>0\) and \(J\in\mathbb{N}\) let \(\mathcal{Q}_\beta(J)\) be the set of maps \((\tau,s)\mapsto Q(\tau\mid s)\) from \([0,1]\times[0,1]^J\) to \(\mathbb{R}\) with \(\|Q\|_{C^\beta}\le1\), where, with \(\ell\) the largest integer strictly below \(\beta\), \(\|Q\|_{C^\beta}=\max_{|\alpha|\le\ell}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=\ell}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-\ell}\) (maxima over multi indices on \(\mathbb{R}^{1+J}\)), and with \(\tau\mapsto Q(\tau\mid s)\) nondecreasing for every \(s\). For \(\beta>1\) and \(\underline c\in(0,1]\) let \(\mathcal{Q}_\beta(J;\underline c)\) be the set of \(Q\in\mathcal{Q}_\beta(J)\) with \(\partial_\tau Q\ge\underline c\) on \([0,1]\times[0,1]^J\), so that every \(P_{Q,s}=Q(\cdot\mid s)_\#\mathrm{Unif}[0,1]\) has a density between \(1\) and \(1/\underline c\) on its support and no atom. Let \(\mathscr{W}_{1,\mathbb{R}}\) be the Wasserstein distance of order one on the line, let the training pairs \((S^{(i)},\theta^{(i)})\), \(i\le M\), be i.i.d. with \(S\sim\mu\) and \(\theta\mid S\sim P_{Q,S}\), where \(\mu\) has a Lebesgue density on \([0,1]^J\) between \(c_\mu>0\) and \(C_\mu\), and let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary.
Let \(\beta>2\), \(J\ge1\) and \(\underline c\in(0,1]\). Let \(L_\ell\) be the number of multi indices \(\alpha\) on \(\mathbb{R}^J\) with \(|\alpha|\le\ell\) and \(\Psi(x)=(x^\alpha)_{|\alpha|\le\ell}\in\mathbb{R}^{L_\ell}\). Put \(h=\min\{1/8,M^{-1/(2\beta+J)}\}\), \(I=\{i\le M:\|S^{(i)}-s_0\|_2\le h\}\) and \(X_i=(S^{(i)}-s_0)/h\) for \(i\in I\). For \(\tau\in(0,1)\) let \(\rho_\tau(r)=r(\tau-\mathbf{1}\{r<0\})\) and let \(\hat c_\tau\) be the minimizer of least Euclidean norm of \(c\mapsto\sum_{i\in I}\rho_\tau(\theta^{(i)}-c\cdot\Psi(X_i))\) over \(\mathbb{R}^{L_\ell}\) when the vectors \(\Psi(X_i)\), \(i\in I\), span \(\mathbb{R}^{L_\ell}\), and \(\hat c_\tau=0\) otherwise. Let \(\hat Q(\tau)=\max\{-1,\min\{1,\hat c_\tau\cdot\Psi(0)\}\}\) and \(\hat P_M(\cdot\mid s_0)=\hat Q_\#\mathrm{Unif}[0,1]\). There is a constant \(C\) depending only on \(\beta\), \(J\), \(c_\mu\), \(C_\mu\) and \(\underline c\) such that for every \(M\ge1\) \[\sup_{Q\in\mathcal{Q}_\beta(J;\underline c)}\mathbb{E}\,\mathscr{W}_{1,\mathbb{R}}\big(\hat P_M(\cdot\mid s_0),P_{Q,s_0}\big)\le C\,M^{-\beta/(2\beta+J)}.\] The two maps in the proof of the lower bound (catalog entry no_uniform_guarantee_for_quantile_generators) have \(\partial_\tau Q\ge\frac18\), so with that theorem and the case \(1<\beta\le2\) (catalog entry quantile_generator_exponent_attained_for_beta_between_one_and_two) the exponent \(\beta/(2\beta+J)\) is the minimax exponent over \(\mathcal{Q}_\beta(J;\underline c)\) at an interior point for every \(\beta>1\) when \(\underline c\le\frac18\).
Source: source/chapters/ext/m3b.tex, label m:prop:qnnUB3.
A local polynomial quantile fit is uniformly small when the conditional densities are bounded
Lemma 16.26, page 303 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>1\), \(J\ge1\) and \(\underline c\in(0,1]\) let \(\ell\) be the largest integer strictly below \(\beta\) and let \(Q:[0,1]\times[0,1]^J\to\mathbb{R}\), \((\tau,s)\mapsto Q(\tau\mid s)\), satisfy \(\|Q\|_{C^\beta}\le1\), where \(\|Q\|_{C^\beta}=\max_{|\alpha|\le\ell}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=\ell}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-\ell}\) (maxima over multi indices on \(\mathbb{R}^{1+J}\)), be nondecreasing in \(\tau\) for every \(s\), and satisfy \(\partial_\tau Q\ge\underline c\) on \([0,1]\times[0,1]^J\). Let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary, \(h\in(0,1/8]\), \(\tau\in(0,1)\), \(\bar B=\{x\in\mathbb{R}^J:\|x\|_2\le1\}\) with volume \(\omega_J\), \(F_x\) the distribution function of \(Q(\cdot\mid s_0+hx)_\#\mathrm{Unif}[0,1]\) for \(x\in\bar B\), and \(T^\ell_\tau(x)=\sum_{|\alpha|\le\ell}\partial_s^\alpha Q(\tau\mid s_0)\,h^{|\alpha|}x^\alpha/\alpha!\), the Taylor polynomial of degree \(\ell\) in \(s\) written in the rescaled variable. Let \(L_\ell\) be the number of multi indices \(\alpha\) on \(\mathbb{R}^J\) with \(|\alpha|\le\ell\), \(\Psi(x)=(x^\alpha)_{|\alpha|\le\ell}\), \(c_\ell\in(0,1]\) the largest number with \(\int_{\bar B}u^2\ge c_\ell\,\omega_J\sup_{\bar B}u^2\) for every polynomial \(u\) of degree at most \(\ell\), \(K_\ell\) the smallest number with \(\max_\alpha|c_\alpha|\le K_\ell\sup_{\bar B}|c\cdot\Psi|\) for every \(c\in\mathbb{R}^{L_\ell}\), and \(K_T=J^{\ell/2}/\ell!\).
Let \(p\) be a polynomial of degree at most \(\ell\), \(w=p-T^\ell_\tau\) and \(\delta(x)=F_x(p(x))-\tau\). Let \(g\) be a probability density on \(\bar B\) with \(g\ge g_->0\), and let \(\kappa\ge0\) be such that \(|\int\delta(x)x^\alpha g(x)\,\mathrm{d}x|\le\kappa\) for \(|\alpha|\le\ell\). Put \[c_7=(g_-\omega_Jc_\ell)^{1/2},\qquad\bar\kappa=L_\ell K_\ell\kappa+6K_Th^\beta/\underline c.\] If \(\min\{\tau,1-\tau\}>2\sqrt2\,\bar\kappa/c_7^3\), then \(\sup_{\bar B}|w|\le8\bar\kappa/c_7^4\).
Source: source/chapters/ext/m3b.tex, label m:lem:qpoly.
No two point lower bound gives a smaller exponent for conditional quantile generators
Proposition 16.27, page 304 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. For \(\beta>0\) and \(J\in\mathbb{N}\) let \(\mathcal{Q}_\beta(J)\) be the set of maps \((\tau,s)\mapsto Q(\tau\mid s)\) from \([0,1]\times[0,1]^J\) to \(\mathbb{R}\) with \(\|Q\|_{C^\beta}\le1\), where, with \(\ell\) the largest integer strictly below \(\beta\), \(\|Q\|_{C^\beta}=\max_{|\alpha|\le\ell}\|\partial^\alpha Q\|_\infty+\max_{|\alpha|=\ell}\sup_{z\ne z'}|\partial^\alpha Q(z)-\partial^\alpha Q(z')|/\|z-z'\|_2^{\beta-\ell}\) (maxima over multi indices on \(\mathbb{R}^{1+J}\)), and with \(\tau\mapsto Q(\tau\mid s)\) nondecreasing for every \(s\). Let \(P_{Q,s}=Q(\cdot\mid s)_\#\mathrm{Unif}[0,1]\), \(\mathscr{W}_{1,\mathbb{R}}\) the Wasserstein distance of order one on the line, and let a training pair be \((S,\theta)\) with \(S\sim\mu\) and \(\theta\mid S\sim P_{Q,S}\), where \(\mu\) has a Lebesgue density on \([0,1]^J\) between \(c_\mu>0\) and \(C_\mu\). Let \(s_0\in[0,1]^J\) be at Euclidean distance at least \(1/8\) from the boundary.
Let \(\beta>0\) and \(J\ge1\). There is a constant \(c_8>0\) depending only on \(\beta\) and \(J\) such that for all \(Q_0,Q_1\in\mathcal{Q}_\beta(J)\), with \(P^{(i)}\) the joint law of one pair under \(Q_i\) and \(\Delta=\mathscr{W}_{1,\mathbb{R}}(P_{Q_0,s_0},P_{Q_1,s_0})\), the squared Hellinger distance satisfies \[H^2\big(P^{(1)},P^{(0)}\big)\ge c_8\,c_\mu\,\Delta^{2+J/\beta}.\] Consequently, for every such pair and every \(M\ge1\) some generator \(\hat P_M\) trained on \(M\) i.i.d. pairs has \(\mathbb{E}\,\mathscr{W}_{1,\mathbb{R}}(\hat P_M(\cdot\mid s_0),P_{Q_i,s_0})\le\{2/(c_8c_\mu M)\}^{\beta/(2\beta+J)}\) for \(i=0\) and \(i=1\), so that no lower bound drawn from two maps in \(\mathcal{Q}_\beta(J)\) has an exponent smaller than \(\beta/(2\beta+J)\).
Source: source/chapters/ext/m3b.tex, label m:prop:qnntwo.
Posterior contraction for the initial condition of a PDE under a sparse deep ReLU network (Polson Ročková) prior
Theorem 16.28, page 306 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Consider data \(Z^{(N)}=(Y_i,X_i)_{i\le N}\) with \(Y_i=\mathscr{G}(\theta)(X_i)+\varepsilon_i\), \(X_i\) i.i.d. uniform on \(\mathcal{X}=[0,T]\times\mathbb{T}^d\) and \(\varepsilon_i\sim N(0,I_W)\) i.i.d. and independent of them, where \(\mathscr{G}(\theta)\) is the solution of an evolution equation with initial condition \(\theta:\mathbb{T}^d\to\mathbb{R}\), and let \(P^N_{\theta_0}\) be the law of the data under \(\theta_0\). Let \(\Pi\) be the Polson Ročková spike and slab prior on deep ReLU networks \(\theta:\mathbb{T}^d\to\mathbb{R}\) with their width and sparsity hyperpriors and a depth of order \(\log N\), let \(\Theta_N\) be the associated sieve, let \(p\) be the input dimension of the networks in \(\Theta_N\) (\(p=d\) when networks on \([0,1]^d\) approximate the restriction of \(\theta_0\) to \([0,1]^d\), or \(p=2d\) when every network is composed with the periodic embedding \(x\mapsto(\cos2\pi x_1,\sin2\pi x_1,\dots,\cos2\pi x_d,\sin2\pi x_d)\)), and set \[B_N:=1+\sup_{\theta\in\Theta_N}\big(\|\theta\|_{L^\infty(\mathbb{T}^d)}+\operatorname{Lip}(\theta)\big),\] so that \(\sup_{\theta\in\Theta_N}\|\theta\|_{H^1(\mathbb{T}^d)}\lesssim B_N\). Let \(\theta_0\in C^\alpha(\mathbb{T}^d)\) with \(d/2+1<\alpha<p\). Suppose the forward map satisfies Conditions 2A, 2B and 2E of Nickl (2024) on \(L^\infty\) balls (uniform \(L^\infty\) bounds, \(L^2\) Lipschitz continuity, and the stability estimate \(\|\mathscr{G}(\theta)-\mathscr{G}(\theta_0)\|_{L^2(\mathcal{X})}\ge c\|\theta-\theta_0\|_{H^{-1}}\)). Then with \(\epsilon_N=N^{-\alpha/(2\alpha+p)}\log^\delta N\), \(\delta>1\), there are \(L,L'>0\) with \[\Pi\big(\theta:\|\mathscr{G}(\theta)-\mathscr{G}(\theta_0)\|_{L^2(\mathcal{X})}\le L\epsilon_N\mid Z^{(N)}\big)\to1,\qquad\Pi\big(\theta:\|\theta-\theta_0\|_{H^{-1}}\le L\epsilon_N\mid Z^{(N)}\big)\to1,\] in \(P^N_{\theta_0}\) probability, and, by interpolation between \(H^{-1}\) and \(H^1\) along the sieve, \[\Pi\big(\theta:\|\theta-\theta_0\|_{L^2}\le L'(B_N\epsilon_N)^{1/2}\mid Z^{(N)}\big)\to1\] in \(P^N_{\theta_0}\) probability. The third statement is a rate only when \(B_N\epsilon_N\to0\).
Source: source/chapters/ext/m3c.tex, label m:thm:PR.
Exactness of the one step summary in the linear Gaussian model
Lemma 16.32, page 310 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the linear model \(Y=\Phi\theta+\varepsilon\), \(\varepsilon\sim N(0,\varsigma^2\operatorname{Id}_N)\), with prior \(\theta\sim N(0,\Sigma_0)\), let \(F=\Phi^\top\Phi/\varsigma^2\) and \(\Sigma_N=(F+\Sigma_0^{-1})^{-1}\), and let the one step summary be evaluated at any point \(\bar\theta\): \[S(Y)=\bar\theta+\Sigma_N\Big(\Phi^\top\big(Y-\Phi\bar\theta\big)/\varsigma^2-\Sigma_0^{-1}\bar\theta\Big).\] Then \(S(Y)=\Sigma_N\big(F\theta+\Phi^\top\varepsilon/\varsigma^2\big)\) identically in \(\bar\theta\): the summary is exactly the posterior mean, and in particular does not depend on the linearization point.
Source: source/chapters/ext/m3c.tex, label m:lem:exact.
Residual law under a localized training proposal: prior fraction mean and variance inflation
Theorem 16.33, page 310 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the linear model \(Y=\Phi\theta+\varepsilon\), \(\varepsilon\sim N(0,\varsigma^2\operatorname{Id}_N)\), with prior \(N(0,\Sigma_0)\) on \(D\) retained modes, let \(F=\Phi^\top\Phi/\varsigma^2\), \(\Sigma_N=(F+\Sigma_0^{-1})^{-1}\), and for a point \(\bar\theta\) let \(S(Y)=\bar\theta+\Sigma_N(\Phi^\top(Y-\Phi\bar\theta)/\varsigma^2-\Sigma_0^{-1}\bar\theta)\) (which equals the posterior mean). Let the training parameter be drawn from \(q_c=N(\bar\theta,c^2\Sigma_N)\), independent of \(\varepsilon\), so that \(c\) is the localization radius in posterior standard deviations, let \(R=\theta-S(Y)\), and let \(P=\Sigma_N\Sigma_0^{-1}\) be the prior fraction operator, with eigenvalues \(p_1,\dots,p_D\), and \(s_j=1-p_j\). Then \[\mathbb{E}_{q_c}(R)=P\bar\theta,\qquad\operatorname{Var}_{q_c}(R)=c^2P\Sigma_NP^\top+\Sigma_NF\Sigma_N,\] both without further hypotheses. If in addition \(\Sigma_0\) and \(\Phi^\top\Phi\) commute, so that one orthonormal basis diagonalizes \(P\), \(\Sigma_N\) and \(F\) together, then in that basis \(\operatorname{Var}_{q_c}(R)_j=\Sigma_{N,j}(s_j+c^2p_j^2)\). Consequently: (i) under the prior itself, \(\mathbb{E}(R)=0\) and \(\operatorname{Var}(R)=\Sigma_N\), so the identity head is exact; (ii) under \(q_c\) the residual has the mean \(P\bar\theta\), the prior shrinkage of the proposal center, which does not depend on \(c\) and is of first order in the prior fraction; a head that adds the sample mean of the residuals to the center therefore displaces the GBC posterior by \(P\bar\theta\), whereas a head that centers the residuals, or subtracts \(P\bar\theta\) analytically, does not; (iii) with the mean removed, the head misstates the posterior standard deviation in direction \(j\) by the factor \(\sqrt{s_j+c^2p_j^2}\), so the relative error in posterior standard deviations is \[\beta(c)=\frac1D\sum_{j\le D}\Big|\sqrt{s_j+c^2p_j^2}-1\Big|,\] which is \(O(\max_jp_j)\) as \(c\to0\) and grows like \(c\,\frac1D\sum_jp_j\) for large \(c\).
Source: source/chapters/ext/m3c.tex, label m:thm:locbias.
The localized GBC design (center and rescale) meets the accuracy conditions (G1) and (G2) in the linear Gaussian model
Corollary 16.37, page 315 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. In the linear model \(Y=\Phi\theta+\varepsilon\), \(\varepsilon\sim N(0,\varsigma^2\operatorname{Id}_N)\), with prior \(N(0,\Sigma_0)\), let \(F=\Phi^\top\Phi/\varsigma^2\), \(\Sigma_N=(F+\Sigma_0^{-1})^{-1}\), and suppose \(\Sigma_0\) and \(\Phi^\top\Phi\) commute; let \(P=\Sigma_N\Sigma_0^{-1}\), with eigenvalues \(p_j\), \(s_j=1-p_j\), and for a preliminary estimate \(\bar\theta_N\) let \(S(Y)=\bar\theta_N+\Sigma_N(\Phi^\top(Y-\Phi\bar\theta_N)/\varsigma^2-\Sigma_0^{-1}\bar\theta_N)\), the posterior mean, so that the exact posterior is \(\Pi_N=N(S(Y),\Sigma_N)\). Consider the localized GBC generator: training parameters \(\theta^{(m)}\sim q_c=N(\bar\theta_N,c^2\Sigma_N)\), \(m\le M\), with simulated data \(Y^{(m)}\); an identity head with the residuals \(\theta^{(m)}-S(Y^{(m)})\) centered at their sample mean, whitened by the sample covariance of the centered residuals rescaled by \((s_j+c^2p_j^2)^{-1/2}\) in the eigenbasis of \(P\), and centered at \(S(Y)\) for the observed data, with no residual mean added to the center, no affine head and no importance weighting back to the prior; let \(\widehat\Pi_N\) be the law it produces. Then: (i) (exactness in population) for every \(c>0\) and every center \(\bar\theta_N\), the law the head produces at the observed data is exactly the posterior \(N(S(Y),\Sigma_N)\): localization contributes no bias at all; (ii) (finite \(M\)) with \(M=M_N\) simulations and \(J=J_N\) retained modes, the rescaled sample covariance of the centered residuals equals \(\Sigma_N^{1/2}(\operatorname{Id}+\Delta_M)\Sigma_N^{1/2}\), where \(\|\Delta_M\|_{op}=O_P(\sqrt{J_N/M_N}+J_N/M_N)\) does not depend on \(c\); hence, when \(\Sigma_{N,j}\lesssim\lambda_j/N\) on the retained modes (\(\lambda_j\) the Laplace eigenvalues), as for the heat and Stokes equations, and \(k>d/2\), \(\mathscr{W}_{1,H^{-k-2}}(\widehat\Pi_N,\Pi_N)=O_P\big(N^{-1/2}(\sqrt{J_N/M_N}+J_N/M_N)\big)\), the distance being the Wasserstein distance of order one for the norm \(\|x\|^2=\sum_j\lambda_j^{-k-2}x_j^2\) on the coefficients in the first \(J\) Laplace modes, so the accuracy condition (G1) \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}(\widehat\Pi_N,\Pi_N)\to0\) holds whenever \(J_N/M_N\to0\), at every radius, and the condition (G2) (exponentially small mass off the localization set and moments of every order) holds whenever the exact posterior \(N(S(Y),\Sigma_N)\) satisfies its first clause; (iii) consequently the transfer theorem (catalog entry bvm_for_gbc) applies verbatim to this algorithm, its accuracy assumption being discharged.
Source: source/chapters/ext/m3c.tex, label m:cor:locG.
Gram matrix of the whitened weights of the one step training summary
Lemma 16.41, page 319 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\mathcal{X}=[0,T]\times\Omega\) with \(X\) uniform on \(\mathcal{X}\), let \((e_j)\) be the Laplace eigenbasis of \(\Omega=[0,1]^d\) (the torus), \(P_J\) the orthogonal projection onto \(\operatorname{span}\{e_j:j\le J\}\cong\mathbb{R}^J\) and \(Q_J=\operatorname{Id}-P_J\). Let \(\mathbb{I}_{\bar\vartheta}\) be the linearization at a point \(\bar\vartheta\) of the solution map \(\theta\mapsto u_\theta\), with values in \(\mathbb{R}^{d_W}\) valued functions on \(\mathcal{X}\), let \(\mathcal{D}=\operatorname{diag}(1/\iota_j)\) be the heat (or Stokes) information diagonal, and define \(K'_{\bar\vartheta}\) by \(\operatorname{Id}+K'_{\bar\vartheta}=T^{-1}\mathcal{D}^{1/2}\mathbb{I}^*_{\bar\vartheta}\mathbb{I}_{\bar\vartheta}\mathcal{D}^{1/2}\), that is \(\langle f,(\operatorname{Id}+K'_{\bar\vartheta})g\rangle=\mathbb{E}\langle(\mathbb{I}_{\bar\vartheta}\mathcal{D}^{1/2}f)(X),(\mathbb{I}_{\bar\vartheta}\mathcal{D}^{1/2}g)(X)\rangle_W\). Let \(\mathcal{A}=\operatorname{Id}+K'_{\bar\vartheta}\) be bounded on \(\ell^2\) with \(\mathcal{A}\succeq a\operatorname{Id}\) for some \(a>0\), and write \(B_J=P_J\mathcal{A}^{-1}P_J\). Let the whitened weights be the \(J\times d_W\) matrix \(\phi(x)\) defined by \(\phi(x)^\top v=(\mathbb{I}_{\bar\vartheta}\mathcal{D}^{1/2}P_J\mathcal{A}^{-1}v)(x)\) for \(v\in\mathbb{R}^J\) (equivalently, row \(j\) is \(\iota_j^{1/2}\) times the weight function \(\mathbb{I}_{\bar\vartheta}P_J(T(\mathbb{I}^*_{\bar\vartheta}\mathbb{I}_{\bar\vartheta})^{-1}e_j)\) of the one step summary linearized at \(\bar\vartheta\)), and \(G_\phi=\mathbb{E}\,\phi(X)\phi(X)^\top\) their Gram matrix. Then \[G_\phi=B_J+B_JP_J\mathcal{A}Q_J(Q_J\mathcal{A}Q_J)^{-1}Q_J\mathcal{A}P_JB_J,\] so that \(G_\phi\succeq B_J\succeq\|\mathcal{A}\|^{-1}_{op}\operatorname{Id}\) and \(\|G_\phi-B_J\|_{op}\le a^{-3}\|K'_{\bar\vartheta}Q_J\|^2_{op}\).
Source: source/chapters/ext/m3c.tex, label m:lem:u2pgram.
No sub Gaussian envelope uniform in the dimension for the whole whitened training residual with fresh designs
Proposition 16.42, page 319 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\mathcal{X}=[0,T]\times\Omega\), \(\Omega=[0,1]^d\) the torus, with \(X\) uniform on \(\mathcal{X}\), and let \(\mathbb{I}_\vartheta\) be the linearization at \(\vartheta\) of the solution map \(\theta\mapsto u_\theta\), whose values are \(\mathbb{R}^{d_W}\) valued functions on \(\mathcal{X}\). Let \(\mathcal{D}=\operatorname{diag}(1/\iota_j)\) be the heat (or Stokes) information diagonal, \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\) with \(\lambda_j\) the Laplace eigenvalues, \(\operatorname{Id}+K'_\vartheta=T^{-1}\mathcal{D}^{1/2}\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta\mathcal{D}^{1/2}\), and \(P_J\) the projection onto the first \(J\) eigenfunctions \(e_1,\dots,e_J\). Fix a linearization point \(\bar\vartheta\) determined by the data \(Z^{(N)}\) at which \(\mathcal{A}=\operatorname{Id}+K'_{\bar\vartheta}\) is bounded and invertible, let \(\phi(x)\) be the \(J\times d_W\) matrix with \(\phi(x)^\top v=(\mathbb{I}_{\bar\vartheta}\mathcal{D}^{1/2}P_J\mathcal{A}^{-1}v)(x)\) for \(v\in\mathbb{R}^J\) (rows \(\phi_j(x)^\top\)), \(G_\phi=\mathbb{E}\,\phi(X)\phi(X)^\top\) and \(b_J=\operatorname{ess\,sup}_x\|\phi(x)\|^2_{op}\). Let each simulated data set \(Z^{(m)}\) be drawn at a training parameter \(\theta^{(m)}\) with its own design points \(X^{(m)}_i\), i.i.d. uniform on \(\mathcal{X}\), and its own noise \(\varepsilon^{(m)}_i\sim N(0,I_W)\), independent of each other and of \(\theta^{(m)}\), let the training summary be the one step summary linearized once at \(\bar\vartheta\) with the population information and no prior term, \(\hat S^{(m)}_{N,j}=\langle e_j,\bar\vartheta\rangle+N^{-1}\sum_{i\le N}\langle\iota_j^{-1/2}\phi_j(X^{(m)}_i),Y^{(m)}_i-u_{\bar\vartheta}(X^{(m)}_i)\rangle_W\) for \(j\le J\), and let \(W^{(m)}=\mathcal{D}^{-1/2}\sqrt N(P_J\theta^{(m)}-\hat S^{(m)}_N)\) be the whitened residual. Let the training parameters have any law given \(Z^{(N)}\), and write \(\|\chi\|_{\psi_2\mid Z^{(N)}}=\inf\{K>0:\mathbb{E}[e^{\chi^2/K^2}\mid Z^{(N)}]\le2\}\). Then for every \(N\), every \(J\) and every unit vector \(v\in\mathbb{R}^J\) \[\big\|\langle v,W^{(m)}\rangle\big\|^2_{\psi_2\mid Z^{(N)}}\ge2\operatorname{ess\,sup}_x\|\phi(x)^\top v\|^2_W,\qquad\sup_{\|u\|_2=1}\big\|\langle u,W^{(m)}\rangle\big\|^2_{\psi_2\mid Z^{(N)}}\ge2b_J\ge\frac{2\,\mathrm{tr}\,G_\phi}{d_W}.\] If \(\mathcal{A}\) is bounded and \(\mathcal{A}\succeq a\operatorname{Id}\) for some \(a>0\), then \(\mathrm{tr}\,G_\phi\ge J\|\mathcal{A}\|^{-1}_{op}\). For the heat and Stokes equations, observed from time zero, \(G_\phi=\operatorname{Id}_J\) and \(\|\langle e_J,W^{(m)}\rangle\|^2_{\psi_2\mid Z^{(N)}}\ge2\|e_J\|^2_{L^\infty}/\iota_J\ge2/\iota_J\), which grows like \(\lambda_J\); for the real trigonometric basis the bound is \(4/\iota_J\).
Source: source/chapters/ext/m3c.tex, label m:prop:u2pfalse.
The whitened training residual assumption (U2’) holds under a localized training proposal
Theorem 16.43, page 320 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Setting. Data \(Z^{(N)}\): \(Y_i=u_\theta(X_i)+\varepsilon_i\), \(i\le N\), \(X_i\) i.i.d. uniform on \(\mathcal{X}=[0,T]\times\Omega\), \(\Omega=[0,1]^d\) the torus, \(\varepsilon_i\sim N(0,I_W)\) in \(\mathbb{R}^{d_W}\); \((e_j)\) the Laplace eigenbasis with eigenvalues \(\lambda_j\); \(\theta_0\) the truth and \(P^N_{\theta_0}\) the law of the data; Nickl’s rescaled Gaussian series prior with \(\delta_N=N^{-\gamma/(2\gamma+d)}\), and \(\tilde\delta_N(\xi)=\delta_N^{(\bar\gamma-\xi)/(\bar\gamma+1)}\) with \(\bar\gamma\) the Sobolev index of the localization ball. Let \(\mathbb{I}_\vartheta\) be the linearization of \(\theta\mapsto u_\theta\) at \(\vartheta\), \(\mathcal{D}=\operatorname{diag}(1/\iota_j)\) the heat (or Stokes) information diagonal with \(\iota_j=(1-e^{-2\nu\lambda_jT})/(2\nu\lambda_jT)\), \(\operatorname{Id}+K'_\vartheta=T^{-1}\mathcal{D}^{1/2}\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta\mathcal{D}^{1/2}\), \(P_J\) the projection onto the first \(J\) modes and \(\operatorname{Id}+\Xi=P_J(\operatorname{Id}+K'_{\theta_0})^{-1}P_J\). Let \(\bar\theta_N\) be a preliminary estimate computed from half of the sample with \(\|\bar\theta_N-\theta_0\|_{H^\xi}=O_P(\tilde\delta_N(\xi))\) for \(0\le\xi\le\bar\gamma\), and let \(J=J_N\asymp N^{d/(2\gamma+d)}\). Let the training parameters be \[\theta^{(m)}=\bar\theta_N+N^{-1/2}\mathcal{D}^{1/2}\mathcal{V}_N^{1/2}\xi^{(m)},\qquad m\le M,\] where \(\mathcal{V}_N\) is a positive semidefinite \(J\times J\) matrix determined by \(Z^{(N)}\) with \(\|\mathcal{V}_N\|_{op}=O_P(1)\), and the \(\xi^{(m)}\) are i.i.d. standard normal on \(\mathbb{R}^J\) given \(Z^{(N)}\), conditioned on \(\|\xi^{(m)}\|_2^2\le4J\) when the forward map is nonlinear. Each simulated data set has its own uniform design \(X^{(m)}_i\) and noise \(\varepsilon^{(m)}_i\). With \(\phi(x)\) the \(J\times d_W\) matrix given by \(\phi(x)^\top v=(\mathbb{I}_{\bar\theta_N}\mathcal{D}^{1/2}P_J(\operatorname{Id}+K'_{\bar\theta_N})^{-1}v)(x)\), rows \(\phi_j(x)^\top\), the training summary is \(\hat S^{(m)}_{N,j}=\langle e_j,\bar\theta_N\rangle+N^{-1}\sum_{i\le N}\langle\iota_j^{-1/2}\phi_j(X^{(m)}_i),Y^{(m)}_i-u_{\bar\theta_N}(X^{(m)}_i)\rangle_W\), \(j\le J\) (linearized once at \(\bar\theta_N\), population information, no prior term), and the whitened residuals are \(W^{(m)}=\mathcal{D}^{-1/2}\sqrt N(P_J\theta^{(m)}-\hat S^{(m)}_N)\), used uncentered and unrescaled by the identity skip Gaussian head generator (catalog entry bernstein_von_mises_for_the_identity_skip_quantile_generator). Assumption (U2’) means: \(W^{(m)}=W^{(m)}_{g}+W^{(m)}_{r}\) with pairs i.i.d. given \(Z^{(N)}\), \(\varkappa_N=\sup_{\|v\|_2=1}\|\langle v,W^{(m)}_{g}\rangle\|_{\psi_2\mid Z^{(N)}}=O_P(1)\), where \(\|\chi\|_{\psi_2\mid Z^{(N)}}=\inf\{K>0:\mathbb{E}[e^{\chi^2/K^2}\mid Z^{(N)}]\le2\}\), and, in \(P^N_{\theta_0}\) probability, \(\mathbb{E}[\|W^{(m)}_{r}\|^2_2\mid Z^{(N)}]\to0\) and \(\|\mathbb{E}[W^{(m)}_{g}W^{(m)\top}_{g}\mid Z^{(N)}]-(\operatorname{Id}+\Xi)\|_{op}\to0\).
Conditions on the linearization, with a constant \(a_\star>0\) and, for every \(B>0\), constants \(L_\star\) and \(C_\infty\) that may depend on \(B\), required for all \(\vartheta\) with \(\|\vartheta\|_{H^{\bar\gamma}}\le B\): (I1) \(\operatorname{Id}+K'_{\theta_0}\) is bounded on \(\ell^2\), \(K'_{\theta_0}\) is compact and \(\operatorname{Id}+K'_{\theta_0}\succeq a_\star\operatorname{Id}\); (I2) \(\|(\mathbb{I}_\vartheta-\mathbb{I}_{\theta_0})\mathcal{D}^{1/2}\|_{\ell^2\to L^2}\le L_\star\|\vartheta-\theta_0\|_{H^\zeta}\), where \(\|\mathcal{T}\|_{\ell^2\to L^2}=\sup_{\|g\|_2\le1}(\mathbb{E}\|(\mathcal{T}g)(X)\|^2_W)^{1/2}\) with \(X\) uniform on \(\mathcal{X}\); (I3) \(\|\mathbb{I}_\vartheta h\|_{L^\infty(\mathcal{X})}\le C_\infty\|h\|_{H^\zeta}\) for all \(h\in H^\zeta\).
Then: (a) for the heat and Stokes equations, observed from time zero, with \(\gamma>d+2\), Assumption (U2’) holds with \(\Xi=0\), \(\varkappa_N\le4/\sqrt3\) and \(W^{(m)}_{g}=-\mathbf{1}_{\mathcal{B}_m}N^{-1/2}\sum_i\phi(X^{(m)}_i)\varepsilon^{(m)}_i\), where \(\mathcal{B}_m=\{\|\hat G^{(m)}_\phi-\operatorname{Id}\|_{op}\le1\}\) and \(\hat G^{(m)}_\phi=N^{-1}\sum_{i\le N}\phi(X^{(m)}_i)\phi(X^{(m)}_i)^\top\); (b) for a nonlinear forward map, Assumption (U2’) holds under Conditions 1 and 2 of Nickl (2024) with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\) (\(\zeta>d/2\) the index of Condition 2D), \(\bar\gamma<\gamma-1\), \(\sqrt N\,\tilde\delta_N(0)\tilde\delta_N(\zeta)\to0\), and (I1) to (I3); (c) let \(\Sigma_0\) be positive definite, \(\Sigma_N=(F_{\bar\theta_N}+\Sigma_0^{-1})^{-1}\) with \(F_{\bar\theta_N}\) the realized information on \(J\) modes, and \(a_\star=1\) in case (a); under the hypotheses of (a) or (b), with probability tending to one \(N^{-1}\mathcal{D}^{1/2}F_{\bar\theta_N}\mathcal{D}^{1/2}\succeq(a_\star/12)\operatorname{Id}\), and then \(\mathcal{V}_N=c^2N\mathcal{D}^{-1/2}\Sigma_N\mathcal{D}^{-1/2}\) has \(\|\mathcal{V}_N\|_{op}\le12c^2/a_\star\).
Source: source/chapters/ext/m3c.tex, label m:thm:u2ploc.
Linearization conditions (I1) to (I4) for the reaction diffusion class
Lemma 16.44, page 324 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\Omega=[0,1]^d\) be the torus with \(d\le3\), \(T>0\), \(\mathcal{X}=[0,T]\times\Omega\), and let \(u_\vartheta\) solve \(\partial_tu-\Delta u=f(u)\), \(u(0)=\vartheta\), with \(f\in C^\infty_c(\mathbb{R})\) (the setting of Theorem 1 of Nickl (2024), with integer \(\gamma>2+3d\)), let \(\theta_0\in C^\infty\) and \(K_f=\sup|f'|\), and let \(\mathbb{I}_\vartheta h\) be the solution \(U\) of \(\partial_tU=\Delta U+f'(u_\vartheta)U\), \(U(0)=h\). Let \(\mathcal{D}=\operatorname{diag}(1/\iota_j)\) be the heat information diagonal in the Laplace eigenbasis, \(\operatorname{Id}+K'_\vartheta=T^{-1}\mathcal{D}^{1/2}\mathbb{I}^*_\vartheta\mathbb{I}_\vartheta\mathcal{D}^{1/2}\), and \(\|\mathcal{T}\|_{\ell^2\to L^2}=\sup_{\|g\|_2\le1}(\mathbb{E}|(\mathcal{T}g)(X)|^2)^{1/2}\) with \(X\) uniform on \(\mathcal{X}\). Consider the conditions, with a constant \(a_\star>0\) and, for every \(B>0\), constants \(L_\star\), \(C_\infty\), \(C_Q\) that may depend on \(B\), for all \(\vartheta,\vartheta'\) with \(\|\vartheta\|_{H^{\bar\gamma}},\|\vartheta'\|_{H^{\bar\gamma}}\le B\): (I1) \(\operatorname{Id}+K'_{\theta_0}\) is bounded on \(\ell^2\), \(K'_{\theta_0}\) is compact and \(\operatorname{Id}+K'_{\theta_0}\succeq a_\star\operatorname{Id}\); (I2) \(\|(\mathbb{I}_\vartheta-\mathbb{I}_{\theta_0})\mathcal{D}^{1/2}\|_{\ell^2\to L^2}\le L_\star\|\vartheta-\theta_0\|_{H^\zeta}\); (I3) \(\|\mathbb{I}_\vartheta h\|_{L^\infty(\mathcal{X})}\le C_\infty\|h\|_{H^\zeta}\) for all \(h\in H^\zeta\); (I4) \(\|u_{\vartheta'}-u_\vartheta-\mathbb{I}_\vartheta[\vartheta'-\vartheta]\|_{L^\infty(\mathcal{X})}\le C_Q\|\vartheta'-\vartheta\|^2_{H^\zeta}\). Then (I1) to (I4) hold for every \(\zeta>d/2\) and every \(\bar\gamma>\max(1,\zeta)\), and \(\|\operatorname{Id}+K'_\vartheta\|_{op}\) is bounded by a constant that depends only on \(K_f\) and \(T\).
Source: source/chapters/ext/m3c.tex, label m:lem:rdlin.
Centered and rescaled training residuals for a nonlinear forward map with a reused design
Proposition 16.47, page 327 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Use the setting of the theorem on the whitened residual assumption under a localized proposal (catalog entry whitened_residual_assumption_under_a_localized_proposal): data \(Y_i=u_\theta(X_i)+\varepsilon_i\) on \(\mathcal{X}=[0,T]\times\Omega\) with the observed design \((X_i)_{i\le N}\), a preliminary estimate \(\bar\theta_N\), \(J=J_N\) retained modes with projection \(P_J\), the linearization \(\mathbb{I}_\vartheta\) of \(\theta\mapsto u_\theta\) and the heat (or Stokes) information diagonal \(\mathcal{D}\). Let \(\Sigma_0\) be positive definite, \(\Sigma_N=(F_{\bar\theta_N}+\Sigma_0^{-1})^{-1}\) with \(F_{\bar\theta_N}\) the realized information on \(J\) modes, \(c>0\), and \(\theta^{(m)}=\bar\theta_N+c\Sigma_N^{1/2}\xi^{(m)}\), \(m\le M\), with the \(\xi^{(m)}\) i.i.d. standard normal on \(\mathbb{R}^J\) given the data, conditioned on \(\|\xi^{(m)}\|^2_2\le4J\). Let the simulated data sets reuse the observed design and draw fresh noise, and let \(S^{(m)}\) be their regularized one step summaries on \(J\) modes formed at \(\bar\theta_N\) with this \(\Sigma_N\), \(S^{(m)}=\bar\theta_N+\Sigma_N\big(\mathbb{I}^*_{\bar\theta_N}\big[\sum_ir^{(m)}_i\delta_{X_i}\big]-\Sigma_0^{-1}\bar\theta_N\big)\) with residuals \(r^{(m)}_i=Y^{(m)}_i-u_{\bar\theta_N}(X_i)\), and \(S(Y)\) the same summary of the observed data. Let \(Q\) be the rescaling of the localized GBC algorithm, multiplication by \((s_j+c^2p_j^2)^{-1/2}\) in the eigenbasis of \(P=\Sigma_N\Sigma_0^{-1}\), whose eigenvalues are \(p_j\), with \(s_j=1-p_j\). Then \[\Sigma_N^{-1/2}Q\big(P_J\theta^{(m)}-S^{(m)}\big)=\mu_N+W^{(m)}_{g}+W^{(m)}_{r}\] with \(\mu_N\) determined by \(Z^{(N)}\), and under the hypotheses of part (b) of that theorem (Conditions 1 and 2 of Nickl (2024) with integer \(\gamma>\max(3\zeta+3d/2+2,\eta_0)\) and \(k>\max(\zeta+5d/2,\eta_0)\), \(\bar\gamma<\gamma-1\), \(\sqrt N\,\tilde\delta_N(0)\tilde\delta_N(\zeta)\to0\), and the linearization conditions (I1) to (I3)) and the quadratic remainder condition (I4) \(\|u_{\vartheta'}-u_\vartheta-\mathbb{I}_\vartheta[\vartheta'-\vartheta]\|_{L^\infty(\mathcal{X})}\le C_Q\|\vartheta'-\vartheta\|^2_{H^\zeta}\) on \(H^{\bar\gamma}\) balls, the pairs \((W^{(m)}_{g},W^{(m)}_{r})\) satisfy Assumption (U2’) with \(\operatorname{Id}\) in place of \(\operatorname{Id}+\Xi\) (i.i.d. given the data, \(\sup_{\|v\|_2=1}\|\langle v,W^{(m)}_{g}\rangle\|_{\psi_2\mid Z^{(N)}}=O_P(1)\), \(\mathbb{E}[\|W^{(m)}_{r}\|^2_2\mid Z^{(N)}]\to0\) and \(\|\mathbb{E}[W^{(m)}_{g}W^{(m)\top}_{g}\mid Z^{(N)}]-\operatorname{Id}\|_{op}\to0\) in probability), and \(\max_m\|W^{(m)}_{r}\|_2\to0\) in probability. If \(J_N/M_N\to0\), the head of the localized GBC algorithm, centered at \(S(Y)\), produces a law \(\widehat\Pi_N\) with \(\sqrt N\,\mathscr{W}_{1,H^{-k-2}}\big(\widehat\Pi_N,N(S(Y),\Sigma_N)\big)\to0\) in probability.
Source: source/chapters/ext/m3c.tex, label m:prop:locres.
Chapter 20. Filters, Gates and Local Rules
Exactness of the particle learning step
Proposition 20.5, page 415 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. First stated in Carvalho, Johannes, Lopes and Polson (2010), Particle learning and smoothing, Statistical Science 25, 88 to 106.
Status: proved.
Statement. Consider a state space model with a static parameter \(\theta\), hidden states \(x_t\) and observations \(y_t\), with the Markov structure of such a model: \(y_{t+1}\) depends on the past only through the essential state \(z_t=(x_t,s_t,\theta)\), and \(p(x_{t+1}\mid z_t,y_{1:t+1})=p(x_{t+1}\mid z_t,y_{t+1})\). Suppose there are conditional sufficient statistics \(s_t\), updated deterministically by \(s_{t+1}=S(s_t,x_t,x_{t+1},y_{t+1})\), such that \(p(\theta\mid x_{0:t},y_{1:t})=p(\theta\mid s_t)\). Particle learning updates the essential state in four steps: (1) resample \(z_t\) with weights proportional to the predictive likelihood \(p(y_{t+1}\mid z_t)=\int p(y_{t+1}\mid x_{t+1},\theta)\,p(x_{t+1}\mid x_t,\theta)\,\mathrm{d}x_{t+1}\); (2) propagate \(x_{t+1}\sim p(x_{t+1}\mid z_t,y_{t+1})\); (3) update \(s_{t+1}=S(s_t,x_t,x_{t+1},y_{t+1})\); (4) draw \(\theta\sim p(\theta\mid s_{t+1})\). Let \(z_t\) have the law \(p(z_t\mid y_{1:t})\). Reweight this law by the predictive likelihood \(p(y_{t+1}\mid z_t)\), which is step one at the level of laws, and then apply steps two to four. Then the new essential state \((x_{t+1},s_{t+1},\theta)\) has the law \(p(x_{t+1},s_{t+1},\theta\mid y_{1:t+1})\).
Source: source/chapters/ext/ch14_pl.tex, label prop:pl_exact.
Gamma law and Thorin measure of the shot noise neuron
Theorem 20.9, page 423 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let the input spikes \(t_k\) form a Poisson process of rate \(\lambda>0\) on \(\mathbb{R}\). Attach to the \(k\)th spike a mark \((\tau_k,\kappa_k)\), the marks being independent with a common law \(M\) on \((0,\infty)^2\) and independent of the spike times, and given the marks let the amplitude \(A_k\) be exponential with rate \(\kappa_k\). Let \(V=\sum_{t_k\le t}A_k e^{-(t-t_k)/\tau_k}\) be the stationary potential at time \(t\). Then for every \(\theta>0\), with both sides possibly equal to \(+\infty\), \(-\log\mathbb{E}\,e^{-\theta V}=\lambda\int\tau\log\big(1+\theta/\kappa\big)\,M(\mathrm{d}\tau,\mathrm{d}\kappa)=\int\log\big(1+\theta/y\big)\,U(\mathrm{d}y)\), where \(U\) is the image of the measure \(\lambda\tau\,M(\mathrm{d}\tau,\mathrm{d}\kappa)\) under \((\tau,\kappa)\mapsto\kappa\). If \(\int\log(1+1/y)\,U(\mathrm{d}y)<\infty\), which is the condition that \(U\) be a Thorin measure, then \(V\) is a generalized Gamma convolution with Thorin measure \(U\) and zero drift, that is, \(\mathbb{E}\,e^{-\theta V}=\exp\big(-\int\log(1+\theta/y)\,U(\mathrm{d}y)\big)\) for \(\theta>0\). Otherwise \(V=\infty\) almost surely. In particular, if all marks equal \((\tau,\kappa)\) then \(V\) has the Gamma law with shape \(\lambda\tau\) and rate \(\kappa\), and if \(\tau_k\) and \(\kappa_k\) are independent and \(\kappa_k\) has law \(\nu\) then \(U=\lambda\,\mathbb{E}[\tau_k]\,\nu\).
Source: source/chapters/ch14_filters.tex, label thm:shot_noise_neuron.
Chapter 21. Kolmogorov, Cybernetics and the Thermodynamics of Information
Information bound for recursive self learning
Proposition 21.4, page 433 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: proved.
Statement. Let \(\Theta\) be a target and let \(M_0\) be a model with \(I(\Theta;M_0)=i_0\), where \(I\) denotes mutual information. Suppose \(M_{k+1}\) is computed from \(M_k\) and from fresh randomness independent of \(\Theta\). Then \(I(\Theta;M_k)\le i_0\) for all \(k\). If at each round the update also uses the output of a verifier \(V_k\) that depends on \(\Theta\) through a channel of capacity \(C\) bits per round, so that \(I(\Theta;V_k\mid M_k)\le C\), then \(I(\Theta;M_k)\le i_0+kC\).
Source: source/chapters/ch15_kolmogorov.tex, label prop:recursive_self_learning.
Chapter 22. Open Problems and the Bridge to Hilbert’s Eighth Problem
Open problem: the inferential sixth problem (a program)
Open Problem 1, page 445 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. This is a program, and its first task is to fix the regime. Consider hard spheres in the Boltzmann Grad limit with Gibbs initial data conditioned on a macroscopic density, velocity and temperature profile, observed through the hydrodynamic fields at a later time with noise of standard deviation \(\sigma\). Find a relation between \(\sigma\), the Knudsen number and the number of particles under which the deviation of the particle fields from the solution of the limiting fluid equations is small compared with \(\sigma\), and prove under it a Bernstein von Mises theorem for smooth functionals of the initial macroscopic profile, with constants that do not depend on the number of particles.
Source: source/chapters/ch16_open.tex, label prob:inferential_sixth.
Open problem: quantitative arbitrary functions for mixed phase space systems
Open Problem 2, page 445 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Give quantitative versions of the method of arbitrary functions (for a real random variable \(X\) with a density \(g\) of total variation \(V<\infty\), \(t>0\) and \(a\in[0,1]\), \(|\mathbb{P}(\{tX\}\in[0,a))-a|\le a(1-a)V/(2t)\), where \(\{\cdot\}\) is the fractional part) for mixed phase space systems such as Hénon Heiles, where invariant tori and chaotic regions coexist, with rates in terms of the measure of the chaotic sea.
Source: source/chapters/ch16_open.tex, label prob:poincare.
Open problem: optimal finite de Finetti rate and propagation of chaos for Kac’s model
Open Problem 3, page 445 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Quantitative estimates for chaos and entropic chaos in Kac’s model are known in several forms. Carlen, Carvalho, Le Roux, Loss and Villani (2010) relate propagation of chaos to entropy and show, in particular, that the entropic rate of convergence to equilibrium can be arbitrarily slow. Hauray and Mischler (2014) compare Kac’s chaos, entropic chaos and Fisher information chaos quantitatively, extend the estimates to measures on the Kac spheres, and treat mixtures in the sense of de Finetti, Hewitt and Savage. We do not know whether these estimates are optimal. Find the optimal rate, as a function of the marginal size \(k\) and of \(N\), in the finite de Finetti theorem for exchangeable laws on the sphere \(S^{N-1}(\sqrt N)\) in relative entropy, with examples showing that it cannot be improved, and use it to prove a propagation of chaos theorem for Kac’s model whose dependence on \(N\) is optimal under explicit hypotheses on the initial law.
Source: source/chapters/ch16_open.tex, label prob:axioms.
Open problem: exact samplers for Brownian functionals from Kent’s eigenvalue expansion
Open Problem 4, page 445 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Open problem, posed as a program in the book. Kent’s theorem writes the hitting time of a level \(b\) by a regular one dimensional diffusion, started at a reflecting or entrance boundary \(a\), as \(\sum_{n\geq1}E_n/\lambda_n\), where \(E_1,E_2,\dots\) are independent standard exponential variables and \(0<\lambda_1<\lambda_2<\cdots\) are the eigenvalues of minus the generator on \((a,b)\) with Dirichlet condition at \(b\) and the reflecting or entrance condition at \(a\). For each \(n\), the remainder \(\sum_{k>n}E_k/\lambda_k\) after the first \(n\) terms is a generalized Gamma convolution (a law on \([0,\infty)\) with Laplace transform \(s\mapsto e^{-\phi(s)}\), \(\phi(s)=cs+\int_{(0,\infty)}\log(1+s/y)\,U(\mathrm{d}y)\), where \(c\geq0\) and \(U\) is the Thorin measure) with an explicit Thorin measure. The problem is to identify a class of regular diffusions for which this remainder can be sampled exactly with finite expected cost, to prove that the resulting sampler is exact, and to use it to build a library of exact samplers for Brownian functionals.
Source: source/chapters/ch16_open.tex, label prob:yor.
Open problem: the reciprocal Xi function as a generalized Gamma convolution from the mixture representation
Open Problem 5, page 445 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\Xi(x)=\xi(\frac12+ix)\) and \(\Phi(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\), so that \(\Xi(x)=8\int_0^\infty\Phi(u)\cos(2xu)\,\mathrm{d}u\). The Pólya law is the law of a random variable \(X\) with the symmetric density \(\Phi(|u|)/\int_{\mathbb{R}}\Phi(|v|)\,\mathrm{d}v\); its moment generating function is \(\mathbb{E}e^{sX}=\Xi(-is/2)/\Xi(0)\). By Khintchine’s theorem the Pólya law is the law of \(UZ\) with \(U\) uniform on \([-1,1]\) and \(Z\ge0\) independent of \(U\). The law of \(Z\) is not a generalized Gamma convolution, because \(Z\) has finite exponential moments of every order and a nonconstant generalized Gamma convolution does not. The reciprocal of the moment generating function of \(UZ\) at \(2\sqrt\lambda\) is a different matter. It is \(\Xi(0)/\Xi(i\sqrt\lambda)=\prod_n(1+\lambda/\gamma_n^2)^{-1}\), where \(\pm\gamma_n\) are the zeros of \(\Xi\), and it is the Laplace transform of a generalized Gamma convolution (a law on \([0,\infty)\) with Laplace transform \(\exp\{-as-\int_{(0,\infty)}\log(1+s/t)\,U(\mathrm{d}t)\}\), \(a\ge0\), \(U\) the Thorin measure) if and only if the Riemann hypothesis holds. In that case the Thorin measure is the sum of point masses at the \(\gamma_n^2\). Conversely, the logarithmic derivative of the Laplace transform of a generalized Gamma convolution is analytic outside the negative half line, which forces all the \(\gamma_n^2\) to be positive. Prove that this function is the Laplace transform of a generalized Gamma convolution by using the mixture representation \(X=UZ\) and not the location of the zeros.
Source: source/chapters/ch16_open.tex, label prob:mixtures.
Open problem: scale mixture proof of the monotonicity of Perelman’s entropy on manifolds
Open Problem 6, page 446 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. For a probability density \(u_0\) of finite variance on \(\mathbb{R}^n\) let \(u_\tau\) solve the heat equation \(\partial_\tau u=\Delta u\), and let \(W(\tau)=\tau J(u_\tau)+h(u_\tau)-\frac n2\log(4\pi\tau)-n\), where \(J(u)=\int|\nabla u|^2/u\) is the Fisher information and \(h(u)=-\int u\log u\) the differential entropy; \(W\) is nonincreasing. Ni (2004) proved that the analog of \(W\) for the linear heat equation on a Riemannian manifold is nonincreasing when the Ricci curvature is nonnegative. Extend the scale mixture proof of the monotonicity of \(W\) on \(\mathbb{R}^n\) (for initial densities \(u_0=\int\varphi_v\,G(\mathrm{d}v)\), mixtures of the centered Gaussian densities \(\varphi_v\) with covariance \(vI\) under a mixing law \(G\), for which \(u_\tau=\int\varphi_{v+2\tau}\,G(\mathrm{d}v)\)) to Riemannian manifolds with nonnegative Ricci curvature, interpreting the Ricci term as a correction to the subordinated Brownian motion.
Source: source/chapters/ch16_open.tex, label prob:perelman.
Open problem: a Bayesian answer to Wallstrom’s objection for Nelson’s diffusions
Open Problem 7, page 446 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Formulate a prior on Nelson’s diffusions (diffusions with diffusion constant \(\hbar/2m\) whose drift is determined by the density \(\rho\) and the phase \(S\) of a wave function \(\psi=\sqrt\rho\,e^{iS/\hbar}\)) that does not itself impose the quantization condition, and identify data that determine the circulation of the drift around the nodes of the density, so that the posterior concentrates on drifts satisfying the quantization condition. Position statistics alone fix the density and, through the continuity equation, only the divergence of the current, so the data must carry more. A solution would give a Bayesian answer to Wallstrom’s objection. Wallstrom (1994) observed that the condition that \(\psi\) be single valued, which forces the quantization of angular momentum, does not follow from Nelson’s equations, which only involve \(\rho\) and \(\nabla S\); it must be added by hand.
Source: source/chapters/ch16_open.tex, label prob:quantum.
Open problem: uniform entropy production bound for the Kac walk on conditioned products of normal scale mixtures
Open Problem 8, page 446 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Open problem. Consider the Kac walk with \(N\) particles and total collision rate \(N\): the velocities \(v=(v_1,\dots,v_N)\) lie on the sphere \(S^{N-1}(\sqrt N)=\{\sum_{i\leq N}v_i^2=N\}\), and at rate \(N\) a pair \((i,j)\) is chosen uniformly and an angle \(\theta\) uniformly on \([0,2\pi)\), and the pair is rotated by \(\theta\). Let \(\sigma_N\) be the uniform probability law on the sphere. For a probability density \(F\) with respect to \(\sigma_N\) let \(H_N(F)=\int F\log F\,\mathrm{d}\sigma_N\) be its relative entropy with respect to \(\sigma_N\), and let \(D_N(F)=-\frac{\mathrm{d}}{\mathrm{d}t}H_N(F_t)\big|_{t=0}\) be the rate at which \(H_N\) decreases at time zero along the master equation of the walk, where \(F_t\) is the solution started at \(F_0=F\). For comparison, Villani’s argument gives \(D_N(F)\geq2H_N(F)/(N-1)\) for every \(F\), and Gao, Wang and Xia state in a 2026 preprint that \(2/(N-1)\) is the best constant over all \(F\). For a probability density \(f\) on \(\mathbb{R}\) let \(Z_N(f)=\int\prod_{i\leq N}f(v_i)\,\sigma_N(\mathrm{d}v)\) and \(F^f_N(v)=Z_N(f)^{-1}\prod_{i\leq N}f(v_i)\). Fix \(M>3\) and let \(\mathcal{G}_M\) be the set of bounded probability densities \(f\) on \(\mathbb{R}\) that are normal scale mixtures, \(f(v)=\int(2\pi s)^{-1/2}e^{-v^2/(2s)}\,G(\mathrm{d}s)\) with a mixing law \(G\) on \([0,\infty)\) that is a generalized Gamma convolution (a law on \([0,\infty)\) with Laplace transform \(\exp(-cs-\int_{(0,\infty)}\log(1+s/y)\,U(\mathrm{d}y))\) for some \(c\geq0\) and some measure \(U\) on \((0,\infty)\) with \(\int\log(1+1/y)\,U(\mathrm{d}y)<\infty\), the Thorin measure), and that satisfy \(\int v^2f(v)\,\mathrm{d}v=1\) and \(\int v^4f(v)\,\mathrm{d}v\leq M\). The question is whether there is \(\lambda>0\), depending only on \(M\), such that \(D_N(F^f_N)\geq\lambda H_N(F^f_N)\) for every \(N\geq2\) and every \(f\in\mathcal{G}_M\).
Source: source/chapters/ch16_open.tex, label prob:kinetic.
Open problem: consistent posterior probability of a computable blowup event
Open Problem 11, page 447 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Fix a finite horizon and a computable blowup criterion, for example that a norm of the solution of a Galerkin truncation of the equations exceeds a threshold before the horizon. Construct a prior on initial data and forcing for three dimensional Navier Stokes and a sequence of numerical solvers such that the posterior probability of this event is consistent, that is, converges to its true value as the resolution and the amount of data increase, and quantify how the event approximates finite time blowup of the exact solution. Consistency for the blowup of the exact solution itself, under either hypothesis, would require deciding the regularity of the data near the truth, which is the regularity problem.
Source: source/chapters/ch16_open.tex, label prob:blowup.
Open problem: joint posterior for structure function exponents and the inertial range
Open Problem 12, page 447 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. This is a program. At finite Reynolds number the velocity field is smooth, so a Hölder exponent has to be defined through an inertial range statement, such as the scaling \(S_p(\ell)\propto\ell^{\zeta_p}\) of the structure functions. Develop a joint posterior for the exponents \(\zeta_p\) and the inertial range from measured velocity increments, and use it to ask whether laboratory and numerical data concentrate at, above or below the exponents \(\zeta_p=p/3\) of Kolmogorov and Onsager.
Source: source/chapters/ch16_open.tex, label prob:onsager.
Open problem: computational threshold of posterior samplers for planted clique
Open Problem 13, page 448 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Consider the planted clique problem with a known clique size \(k\) in the graph \(G(n,\frac12)\). The book records that approximate message passing succeeds for \(k\ge(1+\varepsilon)\sqrt{n/e}\), and that from a suitable starting state the Metropolis process cannot find a planted clique of size \(k=n^{1/2-\varepsilon}\) in polynomial time. Take as samplers of the posterior over clique locations either Glauber dynamics on vertex sets of size \(k\), or Langevin dynamics for a continuous relaxation of the clique indicator, in both cases initialized from the prior. Determine whether such samplers have a computational phase transition at \(k=\sqrt{n/e}\), in the sense that they succeed in polynomial time above \((1+\varepsilon)\sqrt{n/e}\) and fail below \((1-\varepsilon)\sqrt{n/e}\).
Source: source/chapters/ch16_open.tex, label prob:phase_transitions.
Open problem: convergence rates for generative Bayesian computation by learned Schrödinger bridges
Open Problem 14, page 448 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. Prove convergence rates for generative Bayesian computation via learned Schrödinger bridges in terms of the entropic regularization, the number of simulations and the smoothness of the posterior. Rates in the sample size are known for the entropic optimal transport cost estimated from samples, by Genevay, Chizat, Bach, Cuturi and Peyré (2019) and by Mena and Niles Weed (2019). The problem concerns the learned bridge as an estimator of the posterior.
Source: source/chapters/ch16_open.tex, label prob:transport.
Open problem: heat dissipation bound for a Bayesian update
Open Problem 15, page 448 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. The Landauer bound bounds the heat dissipated by a physical process in terms of the entropy removed, and the Sagawa Ueda inequality bounds the work in terms of the mutual information acquired. For a Bayesian update from a prior \(\mu_0\) to the posterior \(\mu_y\), prove a lower bound on the heat dissipated by any physical implementation in terms of \(\mathrm{KL}(\mu_y\|\mu_0)\) and the information erased, and compare it with the dissipation of Langevin samplers.
Source: source/chapters/ch16_open.tex, label prob:thermodynamics_of_inference.
Open problem: the Pólya law as a limit of ferromagnetic Ising constructions
Open Problem 16, page 448 of Polson and Sokolov (2026), On Hilbert’s 6th Problem: Probability, Mechanics and Bayesian Computation, edition 1. New in this book.
Status: open.
Statement. This is a program whose goal is the Riemann hypothesis. Let \(\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\), \(\Xi(x)=\xi(\frac12+ix)\) and \(\Phi(u)=\sum_{n\ge1}\big(2\pi^2n^4e^{9u}-3\pi n^2e^{5u}\big)\exp(-\pi n^2e^{4u})\), so that \(\Xi(x)=8\int_0^\infty\Phi(u)\cos(2xu)\,\mathrm{d}u\). The Pólya law is the law with the symmetric density \(\Phi(|u|)/\int_{\mathbb{R}}\Phi(|v|)\,\mathrm{d}v\); its moment generating function is \(\Xi(-is/2)/\Xi(0)\). The de Bruijn Newman constant \(\Lambda\) is the real number such that \(H_t(z)=\int_0^\infty e^{tu^2}\Phi(u)\cos(zu)\,\mathrm{d}u\) has only real zeros if and only if \(t\ge\Lambda\), and the Riemann hypothesis is equivalent to \(\Lambda\le0\). A symmetric random variable \(X\) with \(\mathbb{E}e^{bX^2}<\infty\) for every \(b>0\) has the Lee Yang property if \(z\mapsto\mathbb{E}e^{zX}\) has only purely imaginary zeros; the uniform law on \([-1,1]\) and the Ising spin \(\pm1\) have it. Newman’s theorem (1974): if \(X_1,\dots,X_n\) are independent with the Lee Yang property, \(J_{ij}\ge0\) are such that \(\exp(\sum_{i<j}J_{ij}x_ix_j)\) is integrable against the product law, and \(\mathbb{P}_J\) has density proportional to this exponential with respect to the product law, then \(\sum_ia_iX_i\) under \(\mathbb{P}_J\) has the Lee Yang property for every choice of positive weights \(a_i\). The Riemann hypothesis holds if and only if the Pólya law has the Lee Yang property. Decide whether the Pólya law is the weak limit of a sequence of laws of random variables \(X_n\) obtained by such constructions from Ising spins and uniform laws, with \(\sup_n\mathbb{E}e^{c|X_n|}<\infty\) for every \(c>0\), which is the control that carries the Lee Yang property to the limit by Hurwitz’s theorem. A construction would prove the Riemann hypothesis, and a proof that none exists would show that this route to the hypothesis is closed.
Source: source/chapters/ch16_open.tex, label prob:bridge.
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