# Results by chapter <small>Bayesian Causality</small>

The results of [Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning](https://vsokolov.org/books/bayesian_causality_book/) 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: 20 entries (19 proved, 1 conditional). Results of other authors that the book presents are credited in the [chapter summaries](https://github.com/VadimSokolov/bayesian_causality_book/tree/main/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which the book does not prove); open (stated as an open question); counterexample (a proved counterexample); withdrawn (found wrong after release; the entry says what was wrong and which version corrects it).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 4. Groups, Urns and Information

<a id="average_effect_range_under_case_mix"></a>

### Range of the average effect under a stable odds ratio

Proposition 4.3, page 64 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\psi>0$, let $\gamma$ be the baseline log odds of a unit, with probability law $Q$ over $\gamma$, and let every unit have the logit model $\operatorname{logit}\mathbb{P}(Y=1)=\gamma+\psi t$ for treatment $t\in\{0,1\}$. Write $\tau_Q(\psi)=\mathbb{E}_Q\{\operatorname{expit}(\gamma+\psi)-\operatorname{expit}(\gamma)\}$ for the average effect on the risk difference scale. Then, as $Q$ ranges over all probability laws on $\gamma$, $\{\tau_Q(\psi):Q\}=(0,\tanh(\psi/4)]$, and the supremum is attained only by the point mass at $\gamma=-\psi/2$. Equivalently, an average effect $\tau\in(0,1)$ is generated by the odds ratio $e^{\psi}$ under some case mix if and only if $\psi\ge4\operatorname{artanh}(\tau)$.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:prop:range`.

<a id="separability_of_functionals_on_frechet_class"></a>

### Separability

Theorem 4.8, page 68 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let the outcome space $\mathcal Y$ be finite, let $f:\mathcal Y\times\mathcal Y\to\mathbb{R}$, and let $\mathcal F(F_1,F_0)$ be the Fr\'echet class of joint laws on $\mathcal Y\times\mathcal Y$ with margins $F_1$ and $F_0$. The following are equivalent. (i) $f$ is separable: $f(a,b)=\alpha(a)+\beta(b)$ for some $\alpha,\beta:\mathcal Y\to\mathbb{R}$. (ii) $\int f\,dF$ is constant on $\mathcal F(F_1,F_0)$ for every pair of margins. (iii) $\int f\,dF$ is constant on $\mathcal F(F_1,F_0)$ for one pair of margins with full support.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:thm:sep`.

<a id="identified_estimands_are_functions_of_the_margins"></a>

### Identified estimands are functions of the margins

Corollary 4.9, page 68 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\Psi$ be any functional, linear or not, of the joint law $F$ of the two potential outcomes of a unit drawn from the urn, with margins $(F_1,F_0)$. If $\Psi$ is identified from every design that reveals the margins, that is, if $\Psi(F)=\Psi(F')$ whenever $F$ and $F'$ have the same margins, then $\Psi$ is a function of $(F_1,F_0)$.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:cor:complete`.

<a id="stratified_urns_and_the_reversal"></a>

### Stratified urns and the reversal

Proposition 4.11, page 69 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let the units fall into strata $x$ of sizes $n_x$ with $n=\sum_x n_x$. Let the assignment draw $n_{x,1}$ units of stratum $x$ into treatment and the remaining $n_{x,0}$ into control, uniformly at random, with $n_t=\sum_x n_{x,t}$, and let $\bar Y_{x,t}$ be the mean response of arm $t$ in stratum $x$. Then the finite population effect satisfies $\tau_{\mathrm{fp}}=\sum_x (n_x/n)\,\tau_{\mathrm{fp},x}$, and the collapsed contrast $\sum_x (n_{x,1}/n_1)\bar Y_{x,1}-\sum_x (n_{x,0}/n_0)\bar Y_{x,0}$ is unbiased for $\tau_{\mathrm{fp}}$ for every set of potential outcomes if and only if $n_{x,1}/n_1=n_{x,0}/n_0=n_x/n$ for all $x$, that is, if both arms are allocated in proportion to the stratum sizes.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:prop:strat`.

<a id="posterior_selects_the_group"></a>

### The posterior selects the group

Theorem 4.19, page 74 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\mathcal G$ be a finite collection of groups acting on the baseline space, together with the saturated (heterogeneous) model, each with positive prior probability, a smooth positive prior density on its $d_G$ parameters and a unique interior projection of the true cell laws $F^{\circ}$, at which the Kullback and Leibler divergence has a nonsingular Hessian in those parameters. Let cell $(x,t)$ receive $nw_{x,t}$ independent draws from $F^{\circ}_{x,t}$, and let $D_G=\mathrm{KL}_w(F^{\circ}\,\|\,\mathcal S_G)$ be the weighted divergence from $F^{\circ}$ to the stability class $\mathcal S_G$ of $G$. Then almost surely $\frac1n\log\{\Pi(\mathcal S_G\mid\text{data})/\Pi(\mathcal S_H\mid\text{data})\}\to D_H-D_G$. More precisely, the log posterior odds equal $n(D_H-D_G)-\tfrac12(d_G-d_H)\log n+O_p(\sqrt n)$, and the fluctuation is $O_p(1)$ when both models contain $F^{\circ}$.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:thm:select`.

<a id="effect_transport_bound_by_total_variation"></a>

### Transport of a stable effect across case mixes

Proposition 4.27, page 80 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\psi_G(x)$ be the stratum effect at covariate value $x$ on the scale of a group $G$, with range $R_G=\max_x\psi_G(x)-\min_x\psi_G(x)$, and let $\bar\psi_G(Q)=\sum_x Q_x\psi_G(x)$ be its average over the case mix $Q$. Then for two case mixes $Q$ and $Q'$, $|\bar\psi_G(Q')-\bar\psi_G(Q)|\le R_G\,\|Q-Q'\|_{\mathrm{TV}}\le R_G\sqrt{\tfrac12\mathrm{KL}(Q'\,\|\,Q)}$. If the data are exactly stable on $G$, then $R_G=0$ and the effect transports without change on that scale.

Source: [`book_tex/chapters/m02b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m02b.tex), label `gr:prop:dv`.

## Chapter 5. What Double Machine Learning Assumes

<a id="exact_error_decomposition_of_the_robinson_estimator"></a>

### Exact error decomposition of the partialling out estimator

Lemma 5.3, page 96 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Let $Y=\tau T+g(X)+\varepsilon$ with $\mathbb{E}(\varepsilon\mid X,T)=0$ and $T=m(X)+\nu$ with $\mathbb{E}(\nu\mid X)=0$, so that $\ell(X)=\mathbb{E}(Y\mid X)=\tau m(X)+g(X)$. Let $\hat\ell$ and $\hat m$ be fitted regressions with errors $\delta_\ell=\hat\ell-\ell$ and $\delta_m=\hat m-m$, and let $\hat\tau=\sum_i\hat V_{T,i}\hat V_{Y,i}/\sum_i\hat V_{T,i}^2$ with residuals $\hat V_{T,i}=T_i-\hat m(X_i)$ and $\hat V_{Y,i}=Y_i-\hat\ell(X_i)$. Then for any fits and with no approximation, $\hat\tau-\tau=(S_n+C_n+B_n)/\hat J_n$ with $\hat J_n=\frac1n\sum_i\hat V_{T,i}^2$, where $S_n=\frac1n\sum_i\nu_i\varepsilon_i$, $C_n=\frac1n\sum_i(\tau\nu_i\delta_{m,i}-\nu_i\delta_{\ell,i}-\varepsilon_i\delta_{m,i})$ and $B_n=\frac1n\sum_i(\delta_{m,i}\delta_{\ell,i}-\tau\delta_{m,i}^2)$.

Source: [`book_tex/chapters/m03.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m03.tex), label `dml:lem:exact`.

<a id="two_channels_of_the_dml_bias"></a>

### Two channels

Corollary 5.4, page 98 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Let $Y=\tau T+g(X)+\varepsilon$ with $\mathbb{E}(\varepsilon\mid X,T)=0$ and $T=m(X)+\nu$ with $\mathbb{E}(\nu\mid X)=0$, so that $\ell(X)=\mathbb{E}(Y\mid X)=\tau m(X)+g(X)$. Let $\hat\ell$ and $\hat m$ be fitted regressions with errors $\delta_\ell=\hat\ell-\ell$ and $\delta_m=\hat m-m$, and let $\hat\tau=\sum_i\hat V_{T,i}\hat V_{Y,i}/\sum_i\hat V_{T,i}^2$ with residuals $\hat V_{T,i}=T_i-\hat m(X_i)$ and $\hat V_{Y,i}=Y_i-\hat\ell(X_i)$. Let $J_0=\mathbb{E}\nu^2>0$, assume $\nu$, $\varepsilon$, $\delta_m$ and $\delta_\ell$ have finite second moments, and let the fits for each fold use only observations outside that fold, with $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ the norm and inner product in $L_2$ of the covariate law taken over a fresh draw of $X$. Conditional on the fits, fold by fold, $\mathbb{E}(C_n)=0$, $\mathbb{E}(B_n)=\langle\delta_m,\delta_\ell\rangle-\tau\|\delta_m\|^2$ and $\mathbb{E}(\hat J_n)=J_0+\|\delta_m\|^2$, and when the fits are held fixed as $n$ grows, $\hat\tau-\tau$ converges in probability to $\{\langle\delta_m,\delta_\ell\rangle-\tau\|\delta_m\|^2\}/\{J_0+\|\delta_m\|^2\}$, the bias of $\hat\tau$ at the realized fits. With fold specific errors, the inner product and squared norm are averages over folds weighted by fold size, taken separately in the numerator and the denominator.

Source: [`book_tex/chapters/m03.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m03.tex), label `dml:cor:channels`.

## Chapter 7. Orthogonality Is Not Enough

<a id="sharp_rate_condition_for_dml"></a>

### Sharp rate condition

Theorem 7.1, page 139 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Work in the partially linear model $Y=\tau T+g(X)+\varepsilon$, $T=m(X)+\nu$ with fitting errors $\delta_m$ and $\delta_\ell$ of the fitted treatment and outcome regressions, and with $b=\langle\delta_m,\delta_\ell\rangle-\tau\|\delta_m\|^2$ and $J=\mathbb{E}\nu^2$. Let $\mathbb{E}(\nu\mid X)=0$, $\mathbb{E}(\varepsilon\mid T,X)=0$, $\operatorname{Var}(\nu\mid X)=\sigma_\nu^2>0$, $\mathbb{E}(\varepsilon^2\mid\nu,X)=\sigma_\varepsilon^2$ and $\mathbb{E}\varepsilon^4+\mathbb{E}\nu^4<\infty$. Let the fits be cross fitted with a fixed number $K$ of folds, with $\|\delta_m\|+\|\delta_\ell\|=o_p(1)$ for the fits of every fold, and $\mathbb{E}\delta_m^4\le C\|\delta_m\|^4$ and $\mathbb{E}\delta_\ell^4\le C\|\delta_\ell\|^4$ for a constant $C$. Let $b$ denote the fold weighted average $\sum_k(n_k/n)b^{(k)}$ of its values at the fits trained without fold $k$. If $\sqrt n\,b=o_p(1)$, then $\sqrt n(\hat\tau-\tau)$ converges to a normal law with mean zero and variance $\sigma_\varepsilon^2/\sigma_\nu^2$.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:thm:sharp`.

<a id="alignment_of_errors_under_retained_lasso_shrinkage"></a>

### Alignment under retained shrinkage

Proposition 7.2, page 140 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Suppose the treatment and outcome regressions are linear and sparse on a common support $S$ of size $s$ with agreeing signs, the covariates are sub Gaussian with identity covariance, both regressions are fitted by the lasso with one common penalty $\lambda_{\mathrm L}\asymp\sigma\sqrt{\log p/n}$ and the shrunken fits are kept, and the lasso recovers $S$ exactly (which needs a beta min and an irrepresentability condition), with $p\to\infty$ and $s\log^2p=o(n)$. Then on the event of exact recovery $\langle\delta_m,\delta_\ell\rangle=\lambda_{\mathrm L}^2s\{1+o_p(1)\}$ and $\|\delta_m\|^2=\|\delta_\ell\|^2=\lambda_{\mathrm L}^2s\{1+o_p(1)\}$, where $\delta_m$ and $\delta_\ell$ are the errors of the treatment and outcome fits. Hence the cosine $\rho_\delta=\langle\delta_m,\delta_\ell\rangle/(\|\delta_m\|\|\delta_\ell\|)$ tends to $1$ in probability, and $\sqrt n\langle\delta_m,\delta_\ell\rangle\to0$ if and only if $s=o(\sqrt n/\log p)$.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:prop:alignment`.

<a id="attenuation_representation"></a>

### Attenuation representation

Theorem 7.5, page 142 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: conditional on The hypothesis that the cross channel H_n is negligible relative to the own square channel Q_n is assumed, and the book does not show that any fitting method delivers it. The book's own measurements find the ratio not negligible.

Statement. Work in the partially linear model $Y=\tau T+g(X)+\varepsilon$, $T=m(X)+\nu$ with fitting errors $\delta_m$ and $\delta_\ell$ of the fitted treatment and outcome regressions, and with $b=\langle\delta_m,\delta_\ell\rangle-\tau\|\delta_m\|^2$ and $J=\mathbb{E}\nu^2$. Assume the conditions of the sharp rate condition for cross fitted partialling out (fixed number of folds, moment and fourth moment conditions on the errors) other than $\sqrt n\,b=o_p(1)$. Write $Q_n=\sum_k(n_k/n)\|\delta_m^{(k)}\|^2$ and $H_n=\sum_k(n_k/n)\langle\delta_m^{(k)},\delta_\ell^{(k)}\rangle$, with $\delta_m^{(k)}$ and $\delta_\ell^{(k)}$ the errors of the fits trained without fold $k$. Suppose that $H_n=o_p(Q_n)$ and that $\sqrt n\,Q_n\to\infty$ in probability. Then at the realized fits $\hat\tau-\tau=-\tau(1-\mathcal R)+o_p(1-\mathcal R)$ with $\mathcal R=J/(J+Q_n)$, so that $\hat\tau\approx\tau\mathcal R$.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:thm:atten`.

<a id="cross_fitted_variance_identity"></a>

### Cross fitting: the identity that survives

Proposition 7.7, page 146 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Work in the partially linear model $Y=\tau T+g(X)+\varepsilon$, $T=m(X)+\nu$ with fitting errors $\delta_m$ and $\delta_\ell$ of the fitted treatment and outcome regressions, and with $b=\langle\delta_m,\delta_\ell\rangle-\tau\|\delta_m\|^2$ and $J=\mathbb{E}\nu^2$. Let $\hat\tau$ be the pooled cross fitted estimator with folds $I_1,\dots,I_K$ of sizes $n_k$ and fits $\hat\eta^{(-k)}$ that omit $I_k$, and let $b(\hat\eta^{(-k)})$ be the value of $b$ at those fits. Set $\bar\Delta_n=\sum_k(n_k/n)\,b(\hat\eta^{(-k)})\big/\sum_k(n_k/n)\{J+\|\delta_m^{(k)}\|^2\}$ and $U_n=\hat\tau-\tau-\bar\Delta_n$. If $\hat\tau$ and $\bar\Delta_n$ are square integrable (the second holds when $\|\delta_m\|+\|\delta_\ell\|\le1$ for every fit, since then $|\bar\Delta_n|\le\max(1,|\tau|)/J$), then $\operatorname{Var}(\hat\tau)=\operatorname{Var}(\bar\Delta_n)+2\operatorname{Cov}(\bar\Delta_n,U_n)+\operatorname{Var}(U_n)$.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:prop:cfvar`.

<a id="no_uniformly_valid_interval_of_finite_length"></a>

### No uniformly valid interval of finite length

Proposition 7.8, page 149 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Fix $n$. Let $\mathcal P$ be a class of laws of the partially linear model that contains, for each $j\in(0,\bar j\,]$ and each $\tau\in\mathbb{R}$, a law $P_{j,\tau}$ with $\mathbb{E}\operatorname{Var}(T\mid X)=j$ and effect $\tau$, and suppose that for every $L<\infty$, $\lim_{j\downarrow0}\sup_{|\tau|,|\tau'|\le L}\|P_{j,\tau}^{\otimes n}-P_{j,\tau'}^{\otimes n}\|_{\mathrm{TV}}=0$. Let $\mathcal K_n$ be a confidence set with $\inf_{P\in\mathcal P}\mathbb{P}_P(\tau(P)\in\mathcal K_n)\ge1-\alpha$ for some $\alpha<1$, whose graph is jointly measurable (as it is for any interval with measurable endpoints). Then $\sup_{P\in\mathcal P}\mathbb{E}_P\,\mathrm{Leb}(\mathcal K_n)=\infty$. The Gaussian family $T=m(X)+\nu$, $\nu\sim N(0,j)$, $Y=\tau\nu+h(X)+\varepsilon$, $\varepsilon\sim N(0,\sigma_\varepsilon^2)$ satisfies the condition, with $\|P_{j,\tau}^{\otimes n}-P_{j,\tau'}^{\otimes n}\|_{\mathrm{TV}}\le L\sqrt{nj}/\sigma_\varepsilon$ for $|\tau|,|\tau'|\le L$.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:prop:dufour`.

<a id="complete_shrinkage_bias"></a>

### Complete shrinkage bias

Proposition 7.10, page 152 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. First stated in Polson and Sokolov (2026b), Orthogonality is not enough: double machine learning and the Bayesian alternative, working paper.

Status: proved.

Statement. Let $T=X^\top\gamma+\nu$ and $Y=\tau T+X^\top\beta+\varepsilon$, with $\mathbb{E}X=0$, $\operatorname{Var}(X)=\Sigma$, $\mathbb{E}\nu=0$, $\operatorname{Var}(\nu)=\sigma_\nu^2$, $\nu$ uncorrelated with $X$, and $\mathbb{E}(\varepsilon\mid T,X)=0$ with $\mathbb{E}\varepsilon^2<\infty$. Shrinking $\beta$ entirely to zero and regressing $Y$ on $T$ alone converges in probability, under independent sampling as $n\to\infty$, to $\tau_\infty=\tau+\gamma^\top\Sigma\beta/(\gamma^\top\Sigma\gamma+\sigma_\nu^2)$. If $\Sigma=I_p$ and $\gamma_j=\beta_j=c\ne0$ on a set of size $s$ and zero elsewhere, the bias is $sc^2/(sc^2+\sigma_\nu^2)$, which increases in $s$ and tends to one.

Source: [`book_tex/chapters/m04b.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m04b.tex), label `crit:prop:ric`.

## Chapter 20. The Causal Effect of a Merger

<a id="merger_effect_depends_only_on_demand_and_conduct"></a>

### Identification of the merger effect

Proposition 20.5, page 399 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Consider a market with potential prices $\bm p_t(H)$ under each form of competition $H$, and assume modularity: the intervention $\mathrm{do}(H:=H^{\mathrm{post}})$ replaces $H$ in the equilibrium equation of the structural model and leaves the other structural equations, the parameters $(\beta,\gamma,\theta_2)$ and the exogenous variables unchanged. If the pricing game with post merger conduct $H^{\mathrm{post}}$ has a unique equilibrium (or the equilibrium map applies a fixed selection rule), then the potential prices $\bm p_t(H^{\mathrm{post}})$, and hence the price changes $\Delta\bm p_t=\bm p_t(H^{\mathrm{post}})-\bm p_t$, are functions of the nonlinear demand parameters $\theta_2$, the premerger conduct $H$ and the observed data $(\bm s_t,\bm p_t,\bm x_t)$ alone.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:prop:ident`.

<a id="falsifying_a_conduct_model_under_logit"></a>

### Falsifying a conduct model

Proposition 20.7, page 400 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Under logit demand with price coefficient $\alpha$, price satisfies $p_{jt}=\bm w_{jt}^\top\gamma+g_{H^\star}(\bm s_t)_j/\alpha+\omega_{jt}$, where $H^\star$ is the true conduct, $\bm s_t$ the vector of market shares, $\bm w_{jt}$ cost shifters and $g_H$ the known markup function of conduct $H$ in the shares. Let $\bm z_{jt}$ be instruments with $\mathbb{E}[\omega_{jt}\mid\bm z_{jt}]=0$, where $\bm w_{jt}$ is a function of $\bm z_{jt}$. A candidate $H$ satisfies $\mathbb{E}[p_{jt}-\bm w_{jt}^\top\gamma'-g_H(\bm s_t)_j/\alpha'\mid\bm z_{jt}]=0$ for some $(\gamma',\alpha')$ if and only if there are $\alpha'$ and $\bm\gamma_\Delta$ with $\mathbb{E}[g_{H^\star}(\bm s_t)_j/\alpha-g_H(\bm s_t)_j/\alpha'\mid\bm z_{jt}]=\bm w_{jt}^\top\bm\gamma_\Delta$ almost surely.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:prop:conduct`.

<a id="first_order_flatness_at_zero_heterogeneity"></a>

### First order flatness at zero heterogeneity

Proposition 20.11, page 401 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let consumer $i$ have utility $V_{ij}=\delta_j+\sigma\eta_i\zeta_j$ for product $j$, with $\eta_i$ independent of the other draws, $\mathbb{E}\eta_i=0$, $\mathbb{E}\eta_i^2=1$, and $\zeta_j$ a product characteristic, with an outside good $j=0$ having $\zeta_0=0$. Write $s^0_j$ for the logit shares at $\sigma=0$ and $\bar\zeta=\sum_{k=0}^Js^0_k\zeta_k$. Then at $\sigma=0$, $\partial s_j/\partial\sigma=0$ and $\partial^2s_j/\partial\sigma^2=s^0_j\{(\zeta_j-\bar\zeta)^2-\sum_{k=0}^Js^0_k(\zeta_k-\bar\zeta)^2\}$, and the inverted mean utilities satisfy $\delta(\sigma)=\delta(0)+\tfrac12\sigma^2\bm a+o(\sigma^2)$ with $\bm a=-(\partial\bm s/\partial\delta)^{-1}\partial^2\bm s/\partial\sigma^2$.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:prop:flat`.

<a id="markup_pressure_exceeds_guppi_under_logit"></a>

### Markup pressure versus GUPPI under logit

Lemma 20.13, page 402 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Under logit demand with price coefficient $\alpha$ and premerger Bertrand Nash pricing, let firms $f$ and $n$ merge, with total shares $s_f$ and $s_n$, and let $s_j$ be the share of product $j$ and $p_j$ its price. For a product $j$ of firm $f$, the change in the optimal markup at premerger prices is $\Delta m_j=s_n/\{\alpha(1-s_f)(1-s_f-s_n)\}$ and $p_j\,\mathrm{GUPPI}_j=s_n/\{\alpha(1-s_j)(1-s_n)\}$, where GUPPI is the gross upward pricing pressure index, so $\Delta m_j>p_j\,\mathrm{GUPPI}_j$ whenever $s_n>0$, with equality if and only if $s_n=0$. Products of nonmerging firms have $\Delta m_k=0$.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:lem:pressure`.

<a id="bayes_rule_for_merger_review"></a>

### A Bayes rule for merger review

Proposition 20.15, page 404 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\Delta p/p$ be the merging parties' price effect (the mean, over their products and over markets, of the proportional price change $(p^\ast_{jt}-p_{jt})/p_{jt}$) and $r$ a harm threshold. If challenging a benign merger ($\Delta p/p\le r$) costs $\ell_1$ and clearing a harmful one ($\Delta p/p>r$) costs $\ell_2$, the Bayes rule challenges if and only if $\mathbb{P}(\Delta p/p>r\mid\mathrm{data})>\ell_1/(\ell_1+\ell_2)$. If instead clearing costs $\ell_2(\Delta p/p-r)_+$, the rule challenges if and only if $\ell_2\,\mathbb{E}[(\Delta p/p-r)_+\mid\mathrm{data}]>\ell_1\,\mathbb{P}(\Delta p/p\le r\mid\mathrm{data})$.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:prop:decision`.

<a id="model_averaging_over_conduct_variance"></a>

### Model averaging over conduct

Proposition 20.16, page 404 of Polson and Sokolov (2026), *Bayesian Causality: Models, Accounting and the Bayesian Alternative to Double Machine Learning*, edition 1. New in this book.

Status: proved.

Statement. Let $\varpi_H=\mathbb{P}(H\mid\mathrm{data})$ be the posterior probability of conduct $H$, and for an estimand $e$ let $\mu_H=\mathbb{E}[e\mid\mathrm{data},H]$ and $v_H=\operatorname{Var}(e\mid\mathrm{data},H)$. Then $\mathbb{E}[e\mid\mathrm{data}]=\bar\mu=\sum_H\varpi_H\mu_H$ and $\operatorname{Var}(e\mid\mathrm{data})=\sum_H\varpi_Hv_H+\sum_H\varpi_H(\mu_H-\bar\mu)^2$. A procedure that conditions on a single conduct $\hat H$ and reports $\mu_{\hat H}$ has posterior expected squared error $\operatorname{Var}(e\mid\mathrm{data})+(\mu_{\hat H}-\bar\mu)^2$, and its reported variance $v_{\hat H}$ omits the between conduct term.

Source: [`book_tex/chapters/m15.tex`](https://github.com/VadimSokolov/bayesian_causality_book/blob/v1.0.1/book_tex/chapters/m15.tex), label `blp:prop:bma`.

***

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