Posterior consistency at a KL projection

Let $(\mathsf{X},\mathcal{A},\mu )$ be a sample space with a fixed sigma-finite measure and let $(P_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ be a dominated experiment (Dominated statistical experiments). Let $(\mathrm{\Theta },\mathcal{T},\rho )$ be a metric parameter space with its Borel sigma-algebra (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and supplied finite-valued likelihood densities $p_{\theta }$ (Likelihood densities) that are jointly measurable and normalized relative to $\mu$. Let $P_0$ be a probability measure with supplied density $p_0$ (Probability measures and probability spaces) and let $X_1,X_2,\dots$ be IID with law $P_0$ (Identical distribution and IID families). Let $\mathrm{\Pi }$ be a prior probability on $(\mathrm{\Theta },\mathcal{T})$ (Probability measures and probability spaces). Assume AC for the imported directional KL and likelihood-density conventions. Define $K(\theta )$, $K^{\ast }$, and the nonempty set ${\mathrm{\Theta }}^{\ast }$ as in KL projection sets and relative KL neighborhoods. Assume $K^{\ast }<\mathrm{\infty }$ and, for every $ϵ>0$, the measurable set $\mathrm{\Pi }\{\theta :K(\theta )-K^{\ast }\le ϵ\}>0.$ Fix any selected ${\theta }^{\ast }\in {\mathrm{\Theta }}^{\ast }$ and assume that for $\mathrm{\Pi }$-almost every $\theta$ in each such set, $\log (p_{\theta }\mathrm{/}{p}_{{\theta }^{\ast }})$ is $P_0$-integrable, as required by the denominator lemma. Define the likelihood-tilted measures $Q_{\theta }$ as in Likelihood-tilted measures at a KL projection. Define the posterior by Bayes formula for dominated kernels on finite positive evidence fibres; the denominator lemma below makes those fibres $P_0^n$-almost sure. For each $n\ge 1$, use the product measure $Q_{\theta }^n$ and its $P_0^n$ density from Product likelihood tilts have the product density. Assume the test integrals below are well-defined. For every $\eta >0$, suppose there are measurable tests ${\varphi }_n:{\mathsf{X}}^n\to [0,1]$ and a constant ${\beta }_{\eta }>0$ such that ${\varphi }_n(X_1,\dots ,X_n)⟶0P_0^{\mathrm{\infty }}\text{-almost surely},{\sup }_{\theta :d(\theta ,{\mathrm{\Theta }}^{\ast })>\eta }{Q}_{\theta }^n(1-{\varphi }_n)\le e^{-n{\beta }_{\eta }}$ for all sufficiently large $n$, where $d(\theta ,{\mathrm{\Theta }}^{\ast })={\inf }_{\vartheta \in {\mathrm{\Theta }}^{\ast }}\rho (\theta ,\vartheta )$. Then $\mathrm{\Pi }\{\theta :d(\theta ,{\mathrm{\Theta }}^{\ast })>\eta \mid X_1,\dots ,X_n\}⟶0P_0^{\mathrm{\infty }}\text{-almost surely}$ for every fixed $\eta >0$. The type-I expectations need not be summable or exponentially small.

This sufficient test criterion allows a nonunique minimizer set and arbitrary sigma-finite likelihood tilts. Summable type-I expectations imply its a.s. type-I premise, but summability is not required. The original weaker assertion with only both test expectations tending to zero is false for this class, as the explicit counterexample Weak tests against likelihood tilts do not ensure almost-sure posterior consistency ↗ shows. That example satisfies finite relative KL support and has a unique minimizer, yet its posterior leaves every smaller discrete neighborhood infinitely often. The repair therefore retains a sufficient test interface rather than using the invalid weak-test amplification in the cited article's Lemma 3.3.