KL projection sets and relative KL neighborhoods

Assume AC as required by the imported directional KL convention Kullback–Leibler divergence. Let $(\mathrm{\Theta },\mathcal{T},\rho )$ be a metric space, where $\mathcal{T}$ is its Borel sigma-algebra. Let $(\mathsf{X},\mathcal{A},\mu )$ be a sample space with a fixed sigma-finite dominating measure. Suppose the supplied finite-valued densities $(\theta ,x)\mapsto p_{\theta }(x)$ and $x\mapsto p_0(x)$ are jointly measurable and normalized relative to $\mu$, and observations are IID from $P_0$. Use the directional convention for Kullback–Leibler divergence: $K(\theta ):=\mathrm{K}\mathrm{L}(P_0\mathrm{\parallel }{P}_{\theta })\mathrm{.}$ Assume $K^{\ast }:={\inf }_{\theta \in \mathrm{\Theta }}K(\theta )<\mathrm{\infty }$ and ${\mathrm{\Theta }}^{\ast }:=\{\theta \in \mathrm{\Theta }:K(\theta )=K^{\ast }\}\mathrm{\ne }\mathrm{\varnothing }\mathrm{.}$ The set ${\mathrm{\Theta }}^{\ast }$ is the KL projection set of the truth onto the model. The distance to the KL projection set is $d(\theta ,{\mathrm{\Theta }}^{\ast }):={\inf }_{\vartheta \in {\mathrm{\Theta }}^{\ast }}\rho (\theta ,\vartheta )\mathrm{.}$ A relative KL neighborhood of ${\mathrm{\Theta }}^{\ast }$ of radius $ϵ>0$ is $B_{ϵ}:=\{\theta \in \mathrm{\Theta }:K(\theta )-K^{\ast }\le ϵ\}\mathrm{.}$

For any selected ${\theta }^{\ast }\in {\mathrm{\Theta }}^{\ast }$ with finite contrast, the relative contrast on $B_{ϵ}$ satisfies $K(\theta )-K^{\ast }=-{\mathbb{E}}_{P_0}\log \frac{p_{\theta }(X)}{p_{{\theta }^{\ast }}(X)}$ whenever the displayed log ratio is integrable. This definition allows ${\mathrm{\Theta }}^{\ast }$ to contain multiple parameters and does not assert that the posterior selects one of them.