Results by chapter On Statistical Methods for Genetics

The results of On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme 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: 58 entries (57 proved, 1 open). 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which the book does not prove); open (stated as an open question); counterexample (a proved counterexample); withdrawn (found wrong after release; the entry says what was wrong and which version corrects it).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 3. The Fundamental Theorem of Natural Selection: From Deterministic to Stochastic, Fisher to Bayes

Moving projection lemma

Lemma 3.2, page 63 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(P_{uv}(t)\) be smooth genotype frequencies at a locus, with no assumption of Hardy Weinberg proportions, let \(m_{uv}\) be fixed Malthusian fitnesses and \(\bar m=\sum_{u,v}P_{uv}m_{uv}\). Let \(A_{uv}=\beta_u+\beta_v\) be the fitted values of the \(L^2(P(t))\) projection of \(m\) onto the span of the gene contents, with average effects \(\beta_u\), and let \(\delta_{uv}=m_{uv}-A_{uv}\) be the residuals. Then \(\frac{d\bar m}{dt}=\sum_{u,v}\dot P_{uv}A_{uv}+\sum_{u,v}\dot P_{uv}\delta_{uv}=\mathcal{A}+\mathcal{R}\), where \(\mathcal{A}=2\sum_u\dot p_u\beta_u\) is Fisher’s term, with \(p_u\) the frequency of allele \(u\), and, since \(\sum_{u,v}P_{uv}\delta_{uv}=0\), the remainder is \(\mathcal{R}=\sum_{u,v}\dot P_{uv}\delta_{uv}=\langle\delta,\dot P/P\rangle_t\).

Source: chapters/ftns.tex, label ftns:lem:moving.

Whole genome Fundamental Theorem decomposition

Theorem 3.5, page 66 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(P_g(t)\) be genotype frequencies over a finite set of genotypes \(g\), with no assumption of Hardy Weinberg proportions or of linkage equilibrium, let \(m_g\) be the Malthusian fitness of genotype \(g\), let \(\bar m=\sum_g P_g m_g\), let \(V\) be the span of the gene content functions (the additive functions of the genotype), let \(\beta_{ij}\) be the whole genome average effects, defined as the coefficients of the \(L^2(P_t)\) projection of \(m\) onto \(V\), let \(\dot p_{ij}\) be the rates of change of the allele frequencies, and let \(\delta=m-\Pi_t m\) be the residual of that projection. Then \(\frac{d\bar m}{dt}=\mathcal{A}+\mathcal{R}\) with \(\mathcal{A}=2\sum_{i,j}\dot p_{ij}\beta_{ij}\) and \(\mathcal{R}=\sum_g\dot P_g\,\delta_g=\langle\delta,\partial_t\log P\rangle_t\). Moreover \(\mathcal{R}=0\) whenever \(\partial_t\log P\in V\), and if \(\mathcal{R}=0\) for every assignment of Malthusian parameters \(m\) then necessarily \(\partial_t\log P\in V\).

Source: chapters/ftns.tex, label ftns:thm:whole.

Exponential family factorization of the remainder

Theorem 3.8, page 69 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let the genotype distribution move within the family \(\log P_g(t)=\sum_jc_j(t)X_j(g)+\sum_A\theta_A(t)u_A(g)-\psi(c(t),\theta(t))\), where \(\{X_j\}\) spans the space \(V\) of gene contents and \(\{u_A\}\) completes the basis, and let \(\delta\) be the residual of the projection of fitness onto \(V\) in \(L^2(P_t)\). Then \(\langle\delta,\partial\log P/\partial c_j\rangle_t=0\) for every main effect \(j\), and \(\mathcal{R}=\sum_A\langle\delta,u_A\rangle_t\,\dot\theta_A\). Here \(\mathcal{R}=\sum_g\dot P_g\,\delta_g\) is the remainder in \(d\bar m/dt=\mathcal{A}+\mathcal{R}\), with \(\bar m\) the mean Malthusian fitness.

Source: chapters/ftns.tex, label ftns:thm:expfam.

Remainder in Lindley coordinates

Corollary 3.10, page 70 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. At a biallelic locus with all three genotype classes present, parametrize the genotype simplex by \((p,\alpha)\), where \(p\) is the frequency of the allele \(A_1\) and \(\alpha=\log\{4P_{11}P_{22}/P_{12}^2\}\) is Lindley’s Hardy Weinberg disequilibrium coefficient. Let \(m_{11},m_{12},m_{22}\) be the Malthusian fitnesses, \(D=m_{11}-2m_{12}+m_{22}\), and \(\mathcal{I}(\alpha)=(1/P_{11}+4/P_{12}+1/P_{22})^{-1}\) the Fisher information about \(\alpha\) per genotyped individual. Then the remainder of the moving projection lemma, \(\mathcal{R}=\sum_{u,v}\dot P_{uv}\delta_{uv}\) with \(\delta\) the residual of the \(L^2(P)\) projection of \(m\) onto the gene contents, equals \(D\,\mathcal{I}(\alpha)\dot\alpha\).

Source: chapters/ftns.tex, label ftns:thm:lindley.

The residual in closed form at a biallelic locus

Proposition 3.11, page 71 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. At a biallelic locus let the genotypes \(A_1A_1,A_1A_2,A_2A_2\) have frequencies \(P_{11},P_{12},P_{22}\), all positive (all three genotype classes present), and Malthusian fitnesses \(m_{11},m_{12},m_{22}\). Let \(V\) be the span of the gene contents, let \(\delta\) be the residual of the \(L^2(P)\) projection of \(m\) onto \(V\), let \(D=m_{11}-2m_{12}+m_{22}\), let \(\alpha=\log\{4P_{11}P_{22}/P_{12}^2\}\) be Lindley’s Hardy Weinberg disequilibrium coefficient, and let \(\mathcal{I}(\alpha)=(1/P_{11}+4/P_{12}+1/P_{22})^{-1}\) be the Fisher information about \(\alpha\) per genotyped individual. Write \(\sigma^2_G=\sum_{u,v}P_{uv}(m_{uv}-\bar m)^2\) for the total genetic variance, \(\sigma^2_D=\sum_{u,v}P_{uv}\delta_{uv}^2\) for the dominance variance and \(\mathcal{A}\) for the additive variance. Let \(h=(1/P_{11},\,-2/P_{12},\,1/P_{22})\). Then \(h\) spans the orthogonal complement of \(V\) in \(L^2(P)\), and \(\delta=D\,\mathcal{I}(\alpha)\,h\), \(\sigma^2_D=D^2\mathcal{I}(\alpha)\) and \(\mathcal{A}=\sigma^2_G-D^2\mathcal{I}(\alpha)\). For every velocity of the genotype frequencies, the remainder \(\mathcal{R}=\langle\delta,\dot P/P\rangle_t\) equals \(D\,\mathcal{I}(\alpha)\dot\alpha\). Under Hardy Weinberg proportions \(\sigma_D^2=(pqD)^2\), the classical dominance variance \((2pqd)^2\) in the parametrization \(m=(\mu+a,\mu+d,\mu-a)\) with \(D=-2d\).

Source: chapters/ftns.tex, label ftns:prop:resid.

Two locus residual

Proposition 3.14, page 73 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. For two biallelic loci let the haplotypes \(AB,Ab,aB,ab\) have positive frequencies \(x_1,\dots,x_4\) and Malthusian fitnesses \(m_1,\dots,m_4\), let \(n_A,n_B\in\{0,1\}\) be the gene contents and \(V=\operatorname{span}\{1,n_A,n_B\}\), and let \(\delta\) be the residual of the \(L^2(x)\) projection of \(m\) onto \(V\). Define \(\lambda=\log\{x_1x_4/(x_2x_3)\}\), the log odds ratio of the haplotype table, \(E=m_1-m_2-m_3+m_4\), the interaction contrast of fitness, \(\mathcal{I}(\lambda)=(\sum_i1/x_i)^{-1}\), the Fisher information about \(\lambda\) per haplotype, and \(D_{AB}=x_1x_4-x_2x_3\). Let \(h=(1/x_1,-1/x_2,-1/x_3,1/x_4)\). Then \(h\perp V\) in \(L^2(x)\), and \(\delta=E\,\mathcal{I}(\lambda)\,h\), \(\sigma^2_{AB}=E^2\mathcal{I}(\lambda)\) where \(\sigma^2_{AB}=\sum_ix_i\delta_i^2\) is the epistatic variance, and the remainder \(\mathcal{R}=\sum_i\dot x_i\delta_i\) equals \(E\,\mathcal{I}(\lambda)\dot\lambda\). In particular, recombination at rate \(r\) in continuous time, \(\dot x=-rD_{AB}(1,-1,-1,1)\), moves no allele frequency and changes mean fitness at the rate \(\mathcal{R}=-rD_{AB}E\), exactly.

Source: chapters/ftns.tex, label ftns:prop:twolocus.

Exact remainder for one locus with k alleles

Theorem 3.16, page 76 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Consider discrete nonoverlapping generations, one locus with \(k\) alleles, random mating and constant genotypic fitnesses \(w_{ij}\), so that genotype frequencies are \(p_ip_j\). Write \(\bar w=\sum_{i,j}p_ip_jw_{ij}\), \(w_{i\cdot}=\sum_jp_jw_{ij}\), \(\alpha_i=w_{i\cdot}-\bar w\), \(\Delta p_i=p_i\alpha_i/\bar w\), and \(\delta_{ij}=w_{ij}-\bar w-\alpha_i-\alpha_j\). Let \(\mathcal{A}\) denote the additive term, \(\mathcal{A}=2\sum_ip_i\alpha_i^2\). Then \(\Delta\bar w=\frac{\mathcal{A}}{\bar w}+\frac{1}{\bar w^{2}}\sum_{i,j}\delta_{ij}(p_i\alpha_i)(p_j\alpha_j)\). With two alleles and \(D=w_{11}-2w_{12}+w_{22}\) this is \(\Delta\bar w=\mathcal{A}/\bar w+D(\Delta p)^2\).

Source: chapters/ftns.tex, label ftns:thm:main.

Exact relative error of the Fundamental Theorem, two alleles

Corollary 3.17, page 76 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. At a biallelic locus under the conditions of the exact remainder theorem for one locus, with allele frequencies \(p\) and \(q=1-p\), mean fitness \(\bar w\) and \(D=w_{11}-2w_{12}+w_{22}\), the remainder \(\mathcal{R}=D(\Delta p)^2\) satisfies \(\frac{\mathcal{R}}{\mathcal{A}/\bar w}=\frac{D\,pq}{2\bar w}\).

Source: chapters/ftns.tex, label ftns:cor:relerr.

Chapter 4. The Horseshoe and the Fundamental Theorem of Natural Selection

Bias of the plug in estimate of additive variance

Proposition 4.4, page 133 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(y=X\alpha+\varepsilon\) with \(n\) individuals and \(L\) loci, with \(\mathbb{E}\varepsilon=0\) and \(\operatorname{Var}\varepsilon=\sigma_\varepsilon^2I_n\). Suppose the columns of \(X\) are orthogonal, \(x_\ell^\top x_m=0\) for \(\ell\ne m\), with \(x_\ell^\top x_\ell=nh_\ell\), and let \(\hat\alpha^{\mathrm{marg}}_\ell=x_\ell^\top y/(x_\ell^\top x_\ell)\) and \(\sigma_A^2=\sum_\ell h_\ell\alpha_\ell^2\). Then (a) \(\mathbb{E}\big[\sum_\ell h_\ell(\hat\alpha^{\mathrm{marg}}_\ell)^2\big]=\sigma_A^2+L\sigma_\varepsilon^2/n\); (b) if in addition \(\varepsilon\sim N(0,\sigma_\varepsilon^2I_n)\), then \(\operatorname{Var}\big[\sum_\ell h_\ell(\hat\alpha^{\mathrm{marg}}_\ell)^2\big]=4\sigma_\varepsilon^2\sigma_A^2/n+2L\sigma_\varepsilon^4/n^2\).

Source: chapters/hs.tex, label hs:prop:bias.

Shrinkage is conservative

Proposition 4.9, page 139 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\hat\sigma^2_A(\kappa)=\sum_\ell h_\ell(1-\kappa_\ell)^2(\hat\alpha^{\mathrm{marg}}_\ell)^2\) be the additive variance computed from shrunken effects with shrinkage weights \(\kappa_\ell\). For any \(\kappa\in[0,1]^L\), \(\hat\sigma_A^2(\kappa)\le\hat\sigma_A^2(0)=\sum_\ell h_\ell(\hat\alpha^{\mathrm{marg}}_\ell)^2\), with equality only if \(\kappa_\ell=0\) whenever \(\hat\alpha^{\mathrm{marg}}_\ell\ne0\).

Source: chapters/hs.tex, label hs:prop:conservative.

Oracle shrinkage understates the additive variance

Proposition 4.10, page 140 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(y=X\alpha+\varepsilon\) with \(n\) individuals and \(L\) loci, \(\mathbb{E}\varepsilon=0\), \(\operatorname{Var}\varepsilon=\sigma_\varepsilon^2I_n\), and orthogonal columns with \(x_\ell^\top x_\ell=nh_\ell\). Let \(\hat\alpha^{\mathrm{marg}}_\ell=x_\ell^\top y/(x_\ell^\top x_\ell)\), let \(\hat\alpha_\ell=(1-\kappa_\ell)\hat\alpha^{\mathrm{marg}}_\ell\) be the shrunken effect, \(\hat\sigma^2_A(\kappa)=\sum_\ell h_\ell(1-\kappa_\ell)^2(\hat\alpha^{\mathrm{marg}}_\ell)^2\) and \(\sigma_A^2=\sum_\ell h_\ell\alpha_\ell^2\). Let \(v_\ell=\sigma_\varepsilon^2/(nh_\ell)\) and let \(\kappa^*_\ell=v_\ell/(\alpha_\ell^2+v_\ell)\) be the shrinkage weight minimizing \(\mathbb{E}(\hat\alpha_\ell-\alpha_\ell)^2\). Then \(\mathbb{E}[\hat\sigma_A^2(\kappa^*)]=\sum_\ell h_\ell\alpha_\ell^4/(\alpha_\ell^2+v_\ell)<\sigma_A^2\), with shortfall \(\sum_\ell h_\ell\alpha_\ell^2v_\ell/(\alpha_\ell^2+v_\ell)\).

Source: chapters/hs.tex, label hs:prop:oracle.

The coherent estimator is the posterior mean of the quadratic

Proposition 4.11, page 141 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(m_\ell=\mathbb{E}[\alpha_\ell\mid y]\) and \(s_\ell^2=\operatorname{Var}(\alpha_\ell\mid y)\) under the horseshoe model of the chapter. Then \(\mathbb{E}[\sigma_A^2\mid y]=\sum_\ell h_\ell(m_\ell^2+s_\ell^2)\ge\sum_\ell h_\ell m_\ell^2\), the last sum being the plug in of the posterior means.

Source: chapters/hs.tex, label hs:prop:posterior.

Chapter 5. Many Parameter Models in Population Genetics

Inflation of Fisher’s fitted projection

Corollary 5.6, page 167 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(y=X\beta+\varepsilon\) with \(X\) an \(N\times p\) matrix of full column rank \(p\), \(\mathbb{E}\varepsilon=0\) and \(\operatorname{Var}\varepsilon=\sigma^2I_N\), let \(H\) be the orthogonal projection onto the column space of \(X\), and let \(\hat\sigma_A^2=\|Hy\|^2/N\) and \(\sigma_A^2=\|X\beta\|^2/N\). Then \(\mathbb{E}\hat\sigma_A^2=\sigma_A^2+p\sigma^2/N\).

Source: chapters/mp.tex, label mp:prop:optimism.

Saturation of the optimism past the interpolation threshold

Corollary 5.10, page 174 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(y=X\beta+\varepsilon\) with \(X\) an \(N\times p\) matrix, \(\mathbb{E}\varepsilon=0\) and \(\operatorname{Var}\varepsilon=\sigma^2I_N\), let \(H\) be the orthogonal projection onto the column space of \(X\), and let \(\hat\sigma_A^2=\|Hy\|^2/N\) and \(\sigma_A^2=\|X\beta\|^2/N\). When \(H\) has rank \(\min(p,N)\), as it does generically, \(\mathbb{E}\hat\sigma_A^2=\sigma_A^2+\sigma^2\min(p,N)/N\), so the inflation grows linearly in \(p\) up to \(p=N\) and is constant at \(\sigma^2\) thereafter.

Source: chapters/mp.tex, label mp:cor:saturate.

Shrinkage weights are stick breaking weights

Theorem 5.15, page 180 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\kappa\sim\mathrm{Beta}(a,b)\) with \(a\in(0,1]\) and \(b>0\). Then \(\kappa\) has the law of the first stick breaking weight \(W_1\) of \(\mathrm{GEM}(\alpha,\theta)\) under \(\alpha=1-a\) and \(\theta=a+b-1\), and this map is a bijection from \(\{a\in(0,1],\,b>0\}\) onto \(\{0\le\alpha<1,\,\theta>-\alpha\}\).

Source: chapters/mp.tex, label mp:thm:main.

Spectrum and shrinkage profile

Theorem 5.18, page 181 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \((P_i)\) be a random discrete distribution and let \(\nu\) be the mean intensity of its atoms, so that \(\mathbb{E}\sum_ih(P_i)=\int_0^1h(x)\nu(x)\,dx\) for bounded \(h\) vanishing at the origin. Let \(W\) be a size biased pick from \((P_i)\), with density \(f_W\). Then \(f_W(x)=x\,\nu(x)\), equivalently \(\nu(x)=x^{-1}f_W(x)\). Consequently, for a global local prior whose shrinkage weight \(\kappa\sim\mathrm{Beta}(a,b)\) with \(a\in(0,1]\) and \(b>0\) has density \(f_\kappa\), matched by the stick breaking theorem so that \(f_\kappa=f_W\) for the size biased pick of \(\mathrm{PD}(1-a,a+b-1)\), the induced expected allele frequency spectrum in a sample of size \(n\), with \(a_j\) the number of alleles carried by exactly \(j\) sequences, satisfies \(\mathbb{E}\,a_j=\binom{n}{j}\int_0^1x^{j-1}(1-x)^{n-j}f_\kappa(x)\,dx\).

Source: chapters/mp.tex, label mp:thm:spectrum.

Discovery rates of shrinkage priors

Corollary 5.22, page 182 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let a global local prior have shrinkage weight \(\kappa\sim\mathrm{Beta}(a,b)\) with \(a\in(0,1]\) and \(b>0\), matched to \(\mathrm{PD}(1-a,a+b-1)\) by the stick breaking theorem. The induced expected number of occupied blocks in a sample of size \(n\) grows like \(n^{1-a}\) when \(a<1\) and like \((a+b-1)\log n=b\log n\) when \(a=1\). In particular the horseshoe, with \(a=1/2\), discovers of the order of \(n^{1/2}\) signals, and every prior with \(a=1\), that is the uniform shrinkage prior and the ridge like priors \(\mathrm{Beta}(1,b)\), discovers of the order of \(b\log n\).

Source: chapters/mp.tex, label mp:cor:growth.

Effective model size of a shrinkage prior

Theorem 5.23, page 183 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\kappa_\tau=1/(1+\tau^2\lambda^2)\) where \(\lambda\) is such that \(\kappa_1\sim\mathrm{Beta}(a,b)\) with \(a\in(0,1]\) and \(b>0\). Then as \(\tau\downarrow0\), \(\mathbb{E}(1-\kappa_\tau)\sim\frac{\pi}{B(a,b)\sin(\pi a)}\tau^{2a}\) for \(0<a<1\) and \(\mathbb{E}(1-\kappa_\tau)\sim2b\,\tau^2\log(1/\tau)\) for \(a=1\). Consequently, for \(p=n\) coordinates at the global scale \(\tau_n=n^{-1/2}\), with \(m_n\) the sum of the \(1-\kappa_i\), \(\mathbb{E}m_n\sim\frac{\pi}{B(a,b)\sin(\pi a)}n^{1-a}\) for \(0<a<1\) and \(\mathbb{E}m_n\sim b\log n\) for \(a=1\).

Source: chapters/mp.tex, label mp:thm:effsize.

Rates agree between effective model size and number of species

Corollary 5.24, page 183 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. The rates in the effective model size theorem are exactly those of \(\mathbb{E}K_n\), the expected number of blocks in a sample of size \(n\), under \(\mathrm{PD}(\alpha,\theta)\) with \((\alpha,\theta)=(1-a,\,a+b-1)\). The constants agree exactly when \(a=1\), both being \(b\), and for \(a<1\) they differ by the factor \(\Gamma(2-a)\), which lies in \([\Gamma(1.4616),1]\approx[0.8856,1]\) for every \(a\in(0,1]\).

Source: chapters/mp.tex, label mp:cor:agree.

The shrinkage count and the species count fluctuate differently

Proposition 5.27, page 186 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Fix \(a\in(0,1)\) and \(b>0\), let \(\kappa_1,\dots,\kappa_n\) be independent with the law of \(\kappa_{\tau_n}\) at \(\tau_n=n^{-1/2}\), put \(m_n=\sum_{i=1}^n(1-\kappa_i)\), and let \(K_n\) be the number of blocks in a sample of size \(n\) from \(\mathrm{PD}(\alpha,0)\) with \(\alpha=1-a\). Then \(m_n/\mathbb{E}m_n\to1\) in probability, with \(\{\operatorname{Var}m_n\}^{1/2}/\mathbb{E}m_n\sim\frac{\sin(\pi a)}{\pi}\{B(a,2-a)B(a,b)\}^{1/2}n^{-(1-a)/2}\), whereas \(K_n/n^{\alpha}\to M_\alpha\) almost surely, where \(M_\alpha\) is Mittag Leffler with \(\mathbb{E}M_\alpha^r=r!/\Gamma(1+r\alpha)\). The limit is nondegenerate, so the coefficient of variation of \(K_n\) converges to \(\{2\Gamma(1+\alpha)^2/\Gamma(1+2\alpha)-1\}^{1/2}\), equal to \(0.7555\) at \(\alpha=1/2\), while that of \(m_n\) converges to zero.

Source: chapters/mp.tex, label mp:prop:fluct.

Chapter 6. The Ewens Sampling Formula by de Finetti’s Theorem

Exchangeability forces the Ewens predictive weights

Proposition 6.8, page 217 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \((X_i)_{i\ge1}\) be an exchangeable sequence, and suppose there are a diffuse (atomless) distribution function \(H\) on \([0,1]\) with \(X_1\sim H\) and constants \(c_n\in(0,1)\) such that for every \(n\ge1\), almost surely and for all \(x\), \(\mathbb{P}(X_{n+1}\le x\mid X_1,\dots,X_n)=c_nH(x)+(1-c_n)F_n(x)\), where \(F_n\) is the empirical distribution function of \(X_1,\dots,X_n\). Set \(\theta=c_1/(1-c_1)\in(0,\infty)\). Then for every \(n\ge1\), \(c_n=\theta/(\theta+n)\) and \(1-c_n=n/(\theta+n)\), so that \(\mathbb{P}(X_{n+1}\in\cdot\mid X_1,\dots,X_n)=\frac{\theta}{\theta+n}H(\cdot)+\frac{1}{\theta+n}\sum_{i=1}^n\delta_{X_i}(\cdot)\).

Source: chapters/esf.tex, label esf:prop:weights.

Exact finite n variance and a moment estimator of theta

Proposition 6.11, page 219 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \((X_i)_{i\ge1}\) be exchangeable with \(X_1\sim H\), \(H\) a diffuse distribution function on \([0,1]\), and suppose that for constants \(c_n\in(0,1)\) and every \(n\ge1\), \(\mathbb{P}(X_{n+1}\le x\mid X_1,\dots,X_n)=c_nH(x)+(1-c_n)F_n(x)\) almost surely for all \(x\), where \(F_n\) and \(P_n\) are the empirical distribution function and the empirical measure of \(X_1,\dots,X_n\); set \(\theta=c_1/(1-c_1)\). Then for every \(n\ge1\), every \(g\in L^2(H)\) and every \(x\), \(\operatorname{Var}(\int g\,dP_n)=\operatorname{Var}_H(g)\frac{\theta+n}{n(1+\theta)}\) and \(\operatorname{Var}(F_n(x))=H(x)(1-H(x))\frac{\theta+n}{n(1+\theta)}\). Consequently, with \(v_n=\operatorname{Var}(\int g\,dP_n)\), \(\theta=n(\operatorname{Var}_H(g)-v_n)/(nv_n-\operatorname{Var}_H(g))\).

Source: chapters/esf.tex, label esf:prop:exactvar.

The de Finetti architecture is invariant under abundance weighting

Proposition 6.23, page 233 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(P=(p_1\ge p_2\ge\cdots)\) have the Poisson Dirichlet law with parameter \(\theta>0\), let \(\xi_1,\xi_2,\dots\) be independent draws from a diffuse distribution \(H\), and let \(w:(0,1]\to[0,\infty)\) be measurable with \(0<\sum_\ell w(p_\ell)p_\ell<\infty\) almost surely. Given \(P\) and \((\xi_\ell)\), let \(X^w_1,X^w_2,\dots\) be independent with common law \(\tilde P^w=\sum_\ell w(p_\ell)p_\ell\delta_{\xi_\ell}/\sum_\ell w(p_\ell)p_\ell\). Then \((X^w_i)\) is exchangeable, and its empirical distribution functions \(F^w_n\) converge almost surely and uniformly to a random distribution function, the distribution function of \(\tilde P^w\), whose law is the de Finetti measure of the sequence. Relative to the unweighted paintbox only the mixing measure changes.

Source: chapters/esf.tex, label esf:prop:invariant.

The negative branch is not a weighted paintbox

Proposition 6.34, page 239 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\alpha<0\) and \(\theta=m|\alpha|\) with \(m\) a positive integer, so that the two parameter paintbox with parameters \((\alpha,\theta)\) has frequencies \((P_1,\dots,P_m)\sim\mathrm{Dirichlet}(|\alpha|,\dots,|\alpha|)\) on \(m\) types. For no \(\theta'>0\) and no measurable \(w:(0,1]\to[0,\infty)\) with \(0<\sum_\ell w(p_\ell)p_\ell<\infty\) almost surely is the weighted paintbox built on Poisson Dirichlet frequencies \((p_\ell)\) with parameter \(\theta'\), which samples atom \(\ell\) with probability proportional to \(w(p_\ell)p_\ell\), equal in law to the two parameter paintbox with parameters \((\alpha,\theta)\).

Source: chapters/esf.tex, label esf:prop:notweighted.

Chapter 7. Ascertainment and the Two Parameter Poisson Dirichlet Family

Closure of the Ewens Pitman family under weighted sampling

Theorem 7.9, page 269 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\mathrm{PD}(\sigma,\theta)\), with \(0\le\sigma<1\) and \(\theta>-\sigma\), be the two parameter Poisson Dirichlet (Ewens Pitman) law, lay its atoms on \((0,1)\) as the stick breaking subdivision in size biased order, and drop sample points with the density \(f_\alpha(x)=\alpha(1-x)^{\alpha-1}\), \(\alpha>0\), instead of uniformly. The induced law is again a member of the Ewens Pitman family if and only if \(\sigma=0\) or \(\alpha=1\).

Source: chapters/py.tex, label py:thm:pitman.

Rigidity of the residual allocation structure

Theorem 7.11, page 271 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(G\) be continuous and strictly increasing with \(G(0)=0\) and \(G(1)=1\) (equivalently, the sampling density charges every interval), and lay the atoms in size biased order. Then the induced subdivision, read in the order in which its intervals are laid, is again a residual allocation model with independent factors if and only if \(G(r)=r^{\alpha}\) for some \(\alpha>0\). Consequently, for \(\sigma\in(0,1)\) no such sampling density and direction makes the induced subdivision, in its layout order, the stick breaking sequence of another member of the Ewens Pitman family, apart from the uniform density.

Source: chapters/py.tex, label py:thm:rigid.

No Kingman type genealogy produces the two parameter family

Proposition 7.12, page 273 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Consider any genealogical rule in which, with \(j\) branches present, the defining event is a mutation with probability \(p_j\in[0,1]\) and a binary coalescence otherwise, with \(p_j\) depending only on \(j\). Then the number of types satisfies \(K_n=\sum_{j=1}^n\xi_j\) with \(\xi_j\sim\mathrm{Bernoulli}(p_j)\) independent, so \(\operatorname{Var}[K_n]\le\mathbb{E}[K_n]\). No such rule produces \(\mathrm{PD}(\sigma,\theta)\) with \(\sigma>0\), for which \(\operatorname{Var}[K_n]\asymp n^{2\sigma}\gg\mathbb{E}[K_n]\asymp n^{\sigma}\).

Source: chapters/py.tex, label py:prop:nocoal.

Exponent map of the number of types under power sampling

Proposition 7.13, page 275 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(0<\sigma<1\), lay the \(\mathrm{PD}(\sigma,\theta)\) atoms in size biased order, and sample with the density \(f_\alpha(x)=\alpha(1-x)^{\alpha-1}\), \(\alpha>0\). Let \(\widetilde R_k=\prod_{i\le k}(1-B_i)\) be the residual after \(k\) stick breaking factors and \(K_n\) the number of types in a sample of size \(n\). Then almost surely \(\widetilde R_k\,k^{(1-\sigma)/\sigma}\to M\in(0,\infty)\) and \(\log K_n/\log n\to\sigma_*=\sigma/\{\alpha(1-\sigma)+\sigma\}\).

Source: chapters/py.tex, label py:prop:pyimage.

The sigma diversity is a power of the stick envelope

Lemma 7.16, page 279 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(0<\sigma<1\) and \(\theta>-\sigma\), let \(S\) be the almost sure limit of \(K_n/n^{\sigma}\) under \(\mathrm{PD}(\sigma,\theta)\) (the \(\sigma\) diversity), and let \(M=\lim_k\widetilde R_kk^{(1-\sigma)/\sigma}\) be the limit of the scaled stick residuals in size biased order, where \(\widetilde R_k=\prod_{i\le k}(1-B_i)\) and \(B_i\) are the stick breaking fractions. Then \(S=(M/\sigma)^{\sigma}\) almost surely.

Source: chapters/py.tex, label py:lem:MS.

The weighted diversity is a power of the unweighted diversity

Proposition 7.17, page 280 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(0<\sigma<1\) and \(\theta>-\sigma\), lay the \(\mathrm{PD}(\sigma,\theta)\) atoms in size biased order and sample with the density \(f_\alpha(x)=\alpha(1-x)^{\alpha-1}\), \(\alpha>0\) (the weighted paintbox), with \(\sigma_*=\sigma/\{\alpha(1-\sigma)+\sigma\}\). Let \(S_*\) be the almost sure limit of \(K_nn^{-\sigma_*}\) and \(S\) the almost sure limit of \(K_nn^{-\sigma}\) under the unweighted \(\mathrm{PD}(\sigma,\theta)\). Then \(S_*=C\,S^{\kappa}\) almost surely, with \(\kappa=\alpha\sigma_*/\sigma=\alpha/\{\alpha(1-\sigma)+\sigma\}\), for a constant \(C=C(\sigma,\alpha)\).

Source: chapters/py.tex, label py:prop:power.

The constant in the diversity power law

Proposition 7.18, page 280 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. In the identity \(S_*=C\,S^{\kappa}\) between the weighted and unweighted diversities of \(\mathrm{PD}(\sigma,\theta)\) under sampling with the density \(f_\alpha(x)=\alpha(1-x)^{\alpha-1}\) in size biased order, with \(0<\sigma<1\), \(\alpha>0\) and \(\sigma_*=\sigma/\{\alpha(1-\sigma)+\sigma\}\), the constant is \(C=\alpha^{\sigma_*}\sigma^{(\alpha-1)\sigma_*}\,\Gamma(1-\sigma_*)\Gamma(1-\sigma+\sigma_*)/\Gamma(1-\sigma)\). In particular \(C\) does not depend on \(\theta\), and \(C=1\) at \(\alpha=1\).

Source: chapters/py.tex, label py:prop:constant.

Bias of the singleton fraction

Proposition 7.27, page 288 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Suppose the induced frequencies follow the envelope \(p_k=c\,k^{-1/\sigma_*}\) exactly, and write \(A=nc\). Then as \(A\to\infty\), \(\mathbb{E}[A_1\mid P]=\sigma_*\Gamma(1-\sigma_*)A^{\sigma_*}+O(A^{-\delta})\) and \(\mathbb{E}[K_n\mid P]=\Gamma(1-\sigma_*)A^{\sigma_*}-\tfrac12+O(A^{-\delta})\) for some \(\delta>0\), where \(A_1\) is the number of singletons, and consequently \(\mathbb{E}[A_1\mid P]/\mathbb{E}[K_n\mid P]=\sigma_*\big(1+\frac{1}{2\mathbb{E}[K_n\mid P]}\big)+o\big(1/\mathbb{E}[K_n\mid P]\big)\).

Source: chapters/py.tex, label py:prop:rate.

Growth regimes of the number of types under weighted sampling

Theorem 7.33, page 296 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Lay the \(\mathrm{PD}(\sigma,\theta)\) atoms in either layout, \(0<\sigma<1\), and sample with a weight whose distribution function \(G\) is continuous and increasing. (i) If \(G\) is slowly varying with a density \(g\) near the origin such that \(rg(r)\) is slowly varying, then \(\sigma_*=1\) and \(K_n=n/L(n)\) for a slowly varying \(L\). (ii) If \(G\) is regularly varying with index \(\rho\in(0,\infty)\), with a density near the origin regularly varying with index \(\rho-1\), then \(K_n=n^{\sigma_*}L(n)\) with \(\sigma_*=\sigma/\{\rho(1-\sigma)+\sigma\}\). (iii) If \(G(r)=\exp\{-r^{-\tau}(1+o(1))\}\) for some \(\tau>0\), then \(K_n=(\log n)^{\sigma/\{\tau(1-\sigma)\}}L(\log n)\) for a slowly varying \(L\). (iv) If \(G\equiv0\) on \((0,r_0)\) for some \(r_0>0\), then \(K_n\) is bounded almost surely.

Source: chapters/py.tex, label py:thm:regimes.

Chapter 8. Pólya Inverse Gamma Posteriors, Fisher’s Species Model, and Good’s Predictive Rule for the Ewens Sampling Problem

Inverse gamma series for PIG(0)

Proposition 8.7, page 313 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(Y_1,Y_2,\dots\) be independent inverse gamma variables with shape \(3/2\) and scale \(1\), that is \(1/Y_k\sim\mathrm{Ga}(3/2,1)\). Then \(X_0=\sum_{k\ge1}Y_k/(4k^2)\) converges almost surely and satisfies \(\mathbb{E}[e^{-s^2X_0}]=e^{-\gamma s}/\Gamma(1+s)\) for \(s>0\), with \(\gamma\) the Euler Mascheroni constant. In particular \(\mathbb{E}[X_0]=\sum_k1/(2k^2)=\zeta(2)/2=\pi^2/12\), while \(\operatorname{Var}(X_0)=\infty\).

Source: chapters/pig.tex, label pig:prop:pig0.

Good’s rule is the Ewens predictive

Proposition 8.9, page 316 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Under the Ewens sampling formula with parameter \(\theta\) and sample size \(n\), with \(a_r(n)\) the number of species seen exactly \(r\) times, \(\mathbb{E}[a_r(n)]=\frac{\theta}{r}\frac{n!}{(n-r)!}\frac{\Gamma(\theta+n-r)}{\Gamma(\theta+n)}\) for \(1\le r\le n\). Consequently Good’s estimator of the population frequency \(q_r\) of a species seen \(r\) times, \(\mathbb{E}[q_r\mid\text{data}]=\frac{r+1}{n+1}\frac{\mu_{r+1,n+1}}{\mu_{r,n}}\) with \(\mu_{r,n}=\mathbb{E}[a_r(n)]\), equals \(r/(n+\theta)\) for \(r\ge1\), with unseen mass \(q_0=\theta/(n+\theta)\), the probability that the next specimen is a new species.

Source: chapters/pig.tex, label pig:prop:good.

Predictive law of the number of new species

Proposition 8.11, page 317 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Under the Ewens sampling formula, let \(U_m\) be the number of new species among \(m\) further specimens after \(n\) have been seen. Then given \(\theta\), \(U_m\) has the law of \(\sum_{i=n}^{n+m-1}B_i\) with independent \(B_i\sim\mathrm{Bernoulli}(\theta/(\theta+i))\), independent of the first \(n\) specimens, so \(\mathbb{E}[U_m\mid\theta]=\theta[\psi(\theta+n+m)-\psi(\theta+n)]\) and \(\operatorname{Var}[U_m\mid\theta]=\sum_{i=n}^{n+m-1}\theta i/(\theta+i)^2\), where \(\psi\) is the digamma function.

Source: chapters/pig.tex, label pig:prop:newspecies.

Asymptotic redundancy of the Bayes estimator of theta

Proposition 8.13, page 319 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(K\) be the number of species in a sample of \(n\) specimens under the Ewens sampling formula with parameter \(\theta\), let \(\varphi=\log\theta\), let \(\hat\theta\) and \(\hat\varphi=\log\hat\theta\) be the maximum likelihood estimates, let \(J_n=\operatorname{Var}(K\mid\hat\theta)\) be the observed information in \(\varphi\), and let \(I(\theta)=\operatorname{Var}(K\mid\theta)/\theta^2\) be the Fisher information in \(\theta\). For any prior with a continuous positive density at \(\hat\theta\), the posterior of \(\varphi\) is asymptotically \(N(\hat\varphi,J_n^{-1})\) as \(J_n\to\infty\). The posterior mean satisfies \(\mathbb{E}[\theta\mid K]=\hat\theta+O(\hat\theta J_n^{-1})\), the central \(95\%\) credible interval equals the Wald interval \(\hat\theta\pm1.96\,I(\hat\theta)^{-1/2}\) up to \(o(\hat\theta J_n^{-1/2})\), and two priors differing by a smooth reweighting give posteriors within total variation \(O(J_n^{-1/2})\) and posterior means within \(O(\hat\theta J_n^{-1})\).

Source: chapters/pig.tex, label pig:prop:bvm.

Tail of the concentration posterior

Proposition 8.15, page 321 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(K\) be the number of species in a sample of \(n\) specimens under the Ewens sampling formula with parameter \(\theta\). Under the prior \(\pi(\theta)\propto\theta^{a_0-1}e^{-b_0\theta}\) with \(a_0>0\), \(b_0\ge0\), the posterior \(p(\theta\mid K)\propto\theta^{K+a_0-1}e^{-b_0\theta}\Gamma(\theta)/\Gamma(\theta+n)\) has right tail \(p(\theta\mid K)\sim c\,\theta^{K+a_0-1-n}e^{-b_0\theta}\) as \(\theta\to\infty\). If \(b_0>0\) the tail is light and every posterior moment is finite. If \(b_0=0\) the tail is Pareto of index \(n-K-a_0\): the posterior is proper if and only if \(K+a_0<n\), and \(\mathbb{E}[\theta^p\mid K]<\infty\) if and only if \(p<n-K-a_0\). Under the flat prior \((a_0,b_0)=(1,0)\) the posterior is proper if and only if \(K\le n-2\), has finite mean if and only if \(K\le n-3\), and finite variance if and only if \(K\le n-4\).

Source: chapters/pig.tex, label pig:prop:tail.

Chapter 9. Rational Conjugacy: Pochhammer Priors and Gamma Ratios

Residue formula for the normalizing constant of a rational density

Theorem 9.3, page 346 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(R=P/Q\) be rational with poles at \(-r_1,\dots,-r_J\) of multiplicities \(\mu_1,\dots,\mu_J\), where \(r_i\in\mathbb{C}\setminus(-\infty,0]\), and write the Laurent coefficients \(\gamma_{ij}=\frac{1}{(\mu_i-j)!}\frac{d^{\mu_i-j}}{d\alpha^{\mu_i-j}}\{(\alpha+r_i)^{\mu_i}R(\alpha)\}|_{\alpha=-r_i}\), so that \(R(\alpha)=\Pi(\alpha)+\sum_i\sum_{j=1}^{\mu_i}\gamma_{ij}/(\alpha+r_i)^j\) with \(\Pi\) the polynomial part. If \(\deg Q\ge\deg P+2\) then \(\Pi\equiv0\), \(\sum_i\gamma_{i1}=0\), and \(\int_0^\infty R(\alpha)\,d\alpha=-\sum_i\gamma_{i1}\log r_i+\sum_i\sum_{j=2}^{\mu_i}\gamma_{ij}/((j-1)r_i^{\,j-1})\), with the principal logarithm.

Source: chapters/rc.tex, label rc:thm:residue.

Mellin transform of a Pochhammer density

Theorem 9.6, page 347 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \([x]^k=x(x+1)\cdots(x+k-1)\) denote the rising factorial, let \(a,c>0\) and let \(m,d,b\) be nonnegative integers with \(b\ge m+d+1\). Let \(R(\alpha)=\alpha^d[\alpha]^m/[c\alpha+a]^b\), with simple poles at \(-r_i\), \(r_i=(a+i-1)/c\), and partial fraction expansion \(R(\alpha)=\sum_{i=1}^b\gamma_i/(c\alpha+a+i-1)\). On the strip \(-\{d+\mathbf{1}(m\ge1)\}<\operatorname{Re}(s)<b-m-d\) the Mellin transform is \(\mathcal{M}(s)=\int_0^\infty\alpha^{s-1}R(\alpha)\,d\alpha=\frac{\pi}{c\sin(\pi s)}\sum_{i=1}^b\gamma_ir_i^{\,s-1}\). When \(b\ge m+d+2\) the strip contains \(s=1\), and then the normalizing constant is \(C=\mathcal{M}(1)=-c^{-1}\sum_i\gamma_i\log r_i\), \(\mathbb{E}(\alpha^{s-1})=\mathcal{M}(s)/\mathcal{M}(1)\) for the law with density proportional to \(R\), and \(\mathbb{E}(\log\alpha)=\sum_i\gamma_i\log^2r_i/(2\sum_i\gamma_i\log r_i)\).

Source: chapters/rc.tex, label rc:thm:mellin.

Shrinkage rate of the half horseshoe, corrected

Corollary 9.8, page 348 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. Corrects a statement of Wang and Polson (2024), Pochhammer priors for sparse count models, arXiv:2402.09583.

Status: proved.

Statement. Let \([x]^k=x(x+1)\cdots(x+k-1)\), and let \(p(\alpha)\propto[\alpha]^m/[c\alpha+a]^b\) on \(\alpha>0\), with \(m,b\) nonnegative integers, \(b\ge m+2\) and \(a,c>0\) (the Pochhammer prior). Let \(\nu=1+\mathbf{1}(m\ge1)\). Then \(\int_0^\delta p\asymp\delta^{\nu}\) as \(\delta\downarrow0\). The Cesaro risk bound \(R_N\le\varepsilon-N^{-1}\log\int_0^{\sqrt\varepsilon}p\) therefore reads \(R_N\le\varepsilon+\frac{\nu}{2N}\log\frac1\varepsilon+O(N^{-1})\), minimized at \(\varepsilon=\nu/(2N)\) to give \(R_N=\frac{\nu}{2}N^{-1}\log N\{1+o(1)\}\). Hence every \(m\) attains the rate \(N^{-1}\log N\), and \(m=0\) gives the smaller constant, \(1/2\) against \(1\) for every \(m\ge1\).

Supersedes: The reading of Proposition 2 of Wang and Polson (2024) under which the choice \(m=0\) is needed for the Kullback Leibler superefficient rate \(N^{-1}\log N\).

Source: chapters/rc.tex, label rc:cor:rate.

Exact tail index of the Pochhammer law

Corollary 9.7, page 348 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \([x]^k=x(x+1)\cdots(x+k-1)\), let \(a,c>0\), let \(m,d,b\) be nonnegative integers with \(b\ge m+d+2\), and let \(\alpha\) have the density \(p(\alpha)=\alpha^d[\alpha]^m/\{C[c\alpha+a]^b\}\) on \(\alpha>0\), with \(C\) the normalizing constant. Then \(\mathbb{P}(\alpha>x)\sim x^{-(b-m-d-1)}/\{C\,c^b\,(b-m-d-1)\}\) as \(x\to\infty\), so the law is regularly varying with index \(b-m-d-1\) and has exactly \(b-m-d-2\) finite moments. The half horseshoe \(m=0\), \(b=2\), \(d=0\) has index one and no moments.

Source: chapters/rc.tex, label rc:cor:tail.

Conditional exponentiality of the half horseshoe

Theorem 9.9, page 348 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \([x]^k=x(x+1)\cdots(x+k-1)\), let \(a,c>0\), let \(m,d,b\) be nonnegative integers with \(b\ge m+d+2\), and let \(R(\alpha)=\alpha^d[\alpha]^m/[c\alpha+a]^b\), with poles \(-r_i\), \(r_i=(a+i-1)/c\), and partial fraction expansion \(R(\alpha)=\sum_{i=1}^b\gamma_i/(c\alpha+a+i-1)\). Then \(R\) is completely monotone on \((0,\infty)\) if and only if \(m=d=0\). In that case \(R(\alpha)=\int_0^\infty e^{-\alpha\tau}g(\tau)\,d\tau\) with \(g(\tau)=c^{-1}\sum_{i=1}^b\gamma_ie^{-r_i\tau}\ge0\), which is proportional to the density of \(T=E_1+\dots+E_b\) with \(E_i\) independent exponential with rate \(r_i\); equivalently \(\alpha\mid\tau\sim\mathrm{Exp}(\tau)\) with \(p(\tau)=C^{-1}g(\tau)/\tau\), and the normalizing constant is \(C=\int_0^\infty\tau^{-1}g(\tau)\,d\tau=-c^{-1}\sum_i\gamma_i\log r_i\).

Source: chapters/rc.tex, label rc:thm:cm.

Tilted normalizer with exponential integrals

Theorem 9.16, page 353 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\lambda=-\log z>0\), let \(P\) be a polynomial and write \(P(\alpha)/[c\alpha+a]^b=\Pi(\alpha)+\sum_{i=1}^b\gamma_i/(c\alpha+a+i-1)\) with \(\Pi(\alpha)=\sum_k\pi_k\alpha^k\) and poles \(-r_i\), \(r_i=(a+i-1)/c\). Then \(\int_0^\infty z^{\alpha}P(\alpha)/[c\alpha+a]^b\,d\alpha=\sum_kk!\,\pi_k/\lambda^{k+1}+c^{-1}\sum_{i=1}^b\gamma_ie^{\lambda r_i}E_1(\lambda r_i)\), where \(E_1(x)=\int_x^\infty t^{-1}e^{-t}\,dt\). No degree condition is required, and all moments exist.

Source: chapters/rc.tex, label rc:thm:tilt.

Exact dispersion block for Negative Binomial regression

Proposition 9.21, page 358 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(n_i\), \(i=1,\dots,K\), be Negative Binomial counts with dispersion \(\alpha\) and success probabilities \(\pi_i=e^{\psi_i}/(1+e^{\psi_i})\), where \(\psi_i\) is the linear predictor in the regression coefficients \(\beta\), so that \(p(n_i\mid\alpha,\beta)=\frac{[\alpha]^{n_i}}{n_i!}\frac{(e^{\psi_i})^{n_i}}{(1+e^{\psi_i})^{n_i+\alpha}}\) with \([\alpha]^k=\alpha(\alpha+1)\cdots(\alpha+k-1)\). Marginally in the Polya Gamma augmentation variables \(\omega\), \(p(\alpha\mid n,\beta)\propto\prod_{i=1}^K[\alpha]^{n_i}\,z^{\alpha}\,p(\alpha)\) with \(z=\prod_{i=1}^K(1+e^{\psi_i})^{-1}\). Hence, with a Pochhammer prior \(p(\alpha)\propto[\alpha]^m/[c\alpha+a]^b\) and any integer \(b\ge0\) and \(a,c>0\), the conditional law of \(\alpha\) is exactly the tilted Pochhammer law with numerator \([\alpha]^m\prod_i[\alpha]^{n_i}\), tilt \(z\) and \(\lambda=\sum_i\log(1+e^{\psi_i})=-\log z\), and drawing \(\alpha\) from it before redrawing \(\omega\) gives a partially collapsed Gibbs sampler with the correct stationary distribution.

Source: chapters/rc.tex, label rc:prop:dispersion.

Zipf tails of the Negative Binomial Beta Pochhammer hierarchy

Theorem 9.20, page 358 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \([x]^k=x(x+1)\cdots(x+k-1)\), and let \(\alpha\) have the normalized density \(p_\alpha(\alpha)\propto[\alpha]^m/[c\alpha+a]^b\) on \(\alpha>0\), with \(m,b\) nonnegative integers, \(b\ge m+2\) and \(a,c>0\), let \(\pi\mid\alpha\sim\mathrm{Beta}(c\alpha+a,b)\) and \(n\mid\alpha,\pi\sim\mathrm{NB}(\alpha,\pi)\). Then as \(n\to\infty\), \(p(n)\sim K\,n^{-(1+a)}(\log n)^{-(1+\iota)}\) with \(\iota=1+\mathbf{1}(m\ge1)\) and \(K=\Gamma(a)\Gamma(1+\iota)\varpi/(B(a,b)c^{1+\iota})\), where \(\varpi=\lim_{\alpha\downarrow0}\alpha^{-\mathbf{1}(m\ge1)}p_\alpha(\alpha)\).

Source: chapters/rc.tex, label rc:thm:zipf.

Boundary Zipf law of the Yule Simon prior predictive

Proposition 9.22, page 363 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\rho\) have the normalized density \(p_\rho(\rho)\propto1/[c\rho+a]^b\) on \(\rho>0\), with \(b\ge2\) an integer, \(a,c>0\) and \([x]^b=x(x+1)\cdots(x+b-1)\). The Yule Simon prior predictive, \(p(n)=\int\rho B(n,\rho+1)p_\rho(\rho)\,d\rho\), satisfies \(p(n)\sim p_\rho(0)\,n^{-1}(\log n)^{-2}\) as \(n\to\infty\). This is a proper distribution, since \(\sum_nn^{-1}(\log n)^{-2}<\infty\), but it has no moments of any positive order.

Source: chapters/rc.tex, label rc:prop:boundaryzipf.

Chapter 10. Sufficiency Without a Likelihood in Population Genetics

The check loss identifies the posterior

Theorem 10.8, page 394 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(\mathbb{E}|\theta|<\infty\) and suppose that for \(P\) almost every \(y\) the conditional distribution function \(F_y\) of \(\theta\) given \(Y=y\) is continuous and strictly increasing on the interior of its support. Then \(H^\star=\arg\min_H\mathbb{E}_{(\theta,Y)}\mathbb{E}_{U\sim\mathrm{Unif}(0,1)}\rho_U(\theta-H(U,Y))\), with \(\rho_u\) the check loss, over measurable \(H\) exists, is unique up to sets of measure zero in \((u,y)\), and satisfies \(H^\star(u,y)=F_y^{-1}(u)\).

Source: chapters/suff.tex, label suff:thm:check.

A bottleneck that attains the Bayes risk is sufficient

Theorem 10.12, page 399 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \((\theta,Y)\) have a joint law with \(\theta\) real, prior \(\pi\) and \(\mathbb{E}|\theta|<\infty\), let \(P\) be the marginal law of \(Y\), and suppose that for \(P\) almost every \(y\) the conditional distribution function \(F_y\) of \(\theta\) given \(Y=y\) is continuous and strictly increasing on the interior of its support. Let \(\rho_u(z)=z\{u-\mathbf{1}(z<0)\}\) be the check loss, \(\mathcal{R}(H)=\mathbb{E}_{(\theta,Y)}\mathbb{E}_{U\sim\mathrm{Unif}(0,1)}\rho_U(\theta-H(U,Y))\), and \(\mathcal{R}^\star\) its minimum over measurable \(H\). Suppose \(H(u,y)=g(u,T(y))\) for some measurable \(g\) and \(T\), and \(\mathcal{R}(H)=\mathcal{R}^\star\). Then \(T\) is Bayes sufficient for \(\theta\), that is, the conditional law of \(\theta\) given \(Y=y\) equals its conditional law given \(T(Y)=T(y)\) for \(P\) almost every \(y\). If moreover the data space is countable, as it is for allelic partitions and site frequency spectra, then \(T\) is sufficient in the Fisher Neyman sense for the family \(\{p(\cdot\mid\theta):\theta\in\Theta_0\}\) for a set \(\Theta_0\) of prior probability one; and if in addition the prior has full support and \(\theta\mapsto p(y\mid\theta)\) is continuous for every \(y\), then \(T\) is sufficient for the whole family.

Source: chapters/suff.tex, label suff:thm:suff.

Excess risk is a posterior distance

Theorem 10.16, page 402 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(H(u,y)=g(u,T(y))\) with \(u\mapsto g(u,T(y))\) nondecreasing, and write \(\pi_T(\cdot\mid y)\) for the law with quantile function \(u\mapsto g(u,T(y))\). Suppose that for \(P\) almost every \(y\) the posterior has a density \(f_y\), and that there are constants \(0\le\underline\delta\le\bar\delta<\infty\) such that for almost every \((u,y)\) the inequality \(\underline\delta\le f_y\le\bar\delta\) holds on the closed interval with endpoints \(F_y^{-1}(u)\) and \(g(u,T(y))\). Then \(\frac{\underline\delta}{2}\mathbb{E}_YW_2^2(\pi(\cdot\mid Y),\pi_T(\cdot\mid Y))\le\mathcal{R}(H)-\mathcal{R}^\star\le\frac{\bar\delta}{2}\mathbb{E}_YW_2^2(\pi(\cdot\mid Y),\pi_T(\cdot\mid Y))\), where \(W_2\) is the 2 Wasserstein distance on the line.

Source: chapters/suff.tex, label suff:thm:quant.

Converse of the bottleneck sufficiency theorem

Proposition 10.21, page 404 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. If \(T:\mathcal{Y}\to\mathbb{R}^d\) is Bayes sufficient for \(\theta\) then there exists a measurable \(g:(0,1)\times\mathbb{R}^d\to\Theta\) with \(\mathcal{R}(g(\cdot,T(\cdot)))=\mathcal{R}^\star\).

Source: chapters/suff.tex, label suff:prop:converse.

The number of segregating sites is not sufficient for theta

Proposition 10.25, page 412 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Consider \(n\) sequences under Kingman’s coalescent with infinite sites mutation at scaled rate \(\theta\) and known ancestral states. Let \(\ell_i\) be the total length of branches subtending exactly \(i\) leaves, \(L=\sum_i\ell_i\) the total tree length, \(\xi_i\) the number of sites whose derived allele is carried by exactly \(i\) sequences, \(S=\sum_i\xi_i\) the number of segregating sites, \(a_n=\sum_{j=1}^{n-1}j^{-1}\) and \(b_n=\sum_{j=1}^{n-1}j^{-2}\). Conditionally on the tree the \(\xi_i\) are independent \(\mathrm{Poisson}(\theta\ell_i/2)\), so \(S\sim\mathrm{Poisson}(\theta L/2)\) and \(\xi\mid S,\text{tree}\sim\mathrm{Multinomial}(S;\ell/L)\), free of \(\theta\). Unconditionally, let \(e_{n-1}\) be the spectrum with a single site carried by \(n-1\) sequences and \(\phi_n(\theta)=\mathbb{P}(\xi=e_{n-1}\mid S=1,\theta)=\mathbb{E}[\ell_{n-1}e^{-\theta L/2}]/\mathbb{E}[Le^{-\theta L/2}]\). Then for \(n\ge3\), \(\phi_n(0+)=1/\{(n-1)a_n\}\) and \(\phi_n'(0+)=-(a_n-b_n)/\{(n-1)a_n^2\}<0\). In particular \(\phi_n\) is not constant in \(\theta\), and \(S\) is not sufficient for \(\theta\).

Source: chapters/suff.tex, label suff:prop:notsuff.

Chapter 11. Learning to Coalesce

Bernoulli geometric duality between the Ewens block count and segregating sites

Corollary 11.7, page 439 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(S_2,\dots,S_n\) be independent with \(\mathbb{P}(S_k=j)=p_k(1-p_k)^j\) for \(j\ge0\) and \(p_k=(k-1)/(k-1+\theta)\), the numbers of mutations on a genealogy drawn from Kingman’s coalescent while \(k\) ancestral lineages remain under infinite sites mutation at rate \(\theta/2\). Then \(K_n\overset{d}{=}1+\#\{k\in\{2,\dots,n\}:S_k\ge1\}\), where \(K_n\) is the number of blocks under the Ewens sampling formula with the same \(\theta\).

Source: chapters/coal.tex, label coal:cor:duality.

The number of segregating sites is not sufficient for theta, likelihood argument

Proposition 11.12, page 443 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Under Kingman’s coalescent with infinite sites mutation at rate \(\theta/2\), let \(X\) denote the full infinite sites data on \(n\) sequences, \(S(X)\) its number of segregating sites, \(G\) the genealogy and \(L_G\) its total length. Then \(p(X\mid\theta)=(\theta/2)^{S(X)}\int e^{-\theta L_G/2}w(X,G)\,\mu(dG)\), where \(w\) and the genealogy law \(\mu\) are free of \(\theta\). The conditional law of \(X\) given \(S\) depends on \(\theta\) unless the measures \(w(X,G)\mu(dG)\), pushed forward to \(L_G\), are proportional across configurations with equal \(S\). For \(n=3\) and \(n=4\) they are not, so \(S\) is not sufficient for \(\theta\).

Source: chapters/coal.tex, label coal:prop:notsuff.

DeepSets encoders factor through the site frequency spectrum

Theorem 11.16, page 445 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Let \(X\) be the \(n\times S\) binary matrix of infinite sites data with columns \(x_1,\dots,x_S\), one per segregating site, and call an encoder of per site DeepSets form if \(e(X)=\rho\big(\sum_{j=1}^S\phi(x_j)\big)\) with \(\phi\) invariant under permutation of the \(n\) lineages. Every such encoder satisfies \(e(X)=\rho\big(\sum_{i=1}^{n-1}\xi_i\,\tilde\phi(i)\big)\) for some \(\tilde\phi\), where \(\xi_i\) is the number of sites at which exactly \(i\) of the \(n\) sequences carry the derived allele, and is therefore a function of the site frequency spectrum \(\xi\) alone.

Source: chapters/coal.tex, label coal:thm:deepsets.

The spectrum is a strict coarsening of the gene tree

Corollary 11.17, page 445 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. Under the infinite sites model without recombination, the maximal invariant of the data matrix \(X\) under relabelling of lineages and permutation of sites is the gene tree, that is, the unlabelled rooted topology with mutation counts on its edges. The site frequency spectrum \(\xi\) is a strict coarsening of it, already at \(n=4\) sequences with two segregating sites. Consequently no per site DeepSets encoder, at any depth or width, can recover the information in \(X\) that is not already in \(\xi\).

Source: chapters/coal.tex, label coal:cor:coarse.

The haplotype count escapes the site frequency spectrum

Proposition 11.18, page 446 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: proved.

Statement. The haplotype count \(H=H(X)\), the number of distinct rows of the infinite sites data matrix \(X\), is invariant under relabelling of lineages and under permutation of sites, and \(H\) is not a function of the site frequency spectrum \(\xi\).

Source: chapters/coal.tex, label coal:prop:hap.

Chapter 12. Open Problems and the Future

Conjecture, rigidity of Poisson Dirichlet under general ascertainment

Conjecture 12.7, page 467 of Polson (2026), On Statistical Methods for Genetics: Fisher, Ewens and the Bayesian Programme, edition 1. New in this book.

Status: open.

Statement. Let \(0<\sigma<1\) and \(\vartheta>-\sigma\). There is no sampling weight, depending measurably on the atom index and the atom’s position in the subdivision and charging every interval, that carries \(\mathrm{PD}(\sigma,\vartheta)\) to \(\mathrm{PD}(\sigma',\vartheta')\) for any \((\sigma',\vartheta')\ne(\sigma,\vartheta)\).

Source: chapters/future.tex, label future:conj:rigid.

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