Posterior credibility and frequentist coverage

Let $(\mathrm{\Theta },\mathcal{T})$ be a measurable parameter space and, for each integer $n\ge 1$, let ${\mathcal{E}}_n=(({\mathsf{Y}}_n,{\mathcal{Y}}_n),(P_{\theta ,n}{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment. Let ${\mathrm{\Pi }}_{0,n}$ be a proper prior on $(\mathrm{\Theta },\mathcal{T})$ and let $y\mapsto {\mathrm{\Pi }}_n(\cdot \mid y)$ be a supplied posterior probability kernel under the associated prior-predictive law. For $0<\gamma <1$, a data-indexed set rule $C_n:{\mathsf{Y}}_n\to \mathcal{P}(\mathrm{\Theta })$ is a posterior credible set at level $1-\gamma$ if its membership graph $\{(y,\theta ):\theta \in C_n(y)\}$ is in ${\mathcal{Y}}_n\otimes \mathcal{T}$ and ${\mathrm{\Pi }}_n(C_n(y)\mid y)\ge 1-\gamma$ for every $y$ in its declared data domain. At a fixed truth $\theta \in \mathrm{\Theta }$, its frequentist coverage is $c_n(\theta ):=P_{\theta ,n}\{y:\theta \in C_n(y)\}$. For a truth class ${\mathcal{T}}_n\subseteq \mathrm{\Theta }$, the rule is asymptotically honest at level $1-\gamma$ on ${\mathcal{T}}_n$ if ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{\inf }_{\theta \in {\mathcal{T}}_n}{c}_n(\theta )\ge 1-\gamma$, and it is asymptotically exactly calibrated there if ${\sup }_{\theta \in {\mathcal{T}}_n}\mathrm{\mid }{c}_n(\theta )-(1-\gamma )\mathrm{\mid }\to 0$.

A graph-measurable set has posterior mass at least its declared level under a supplied posterior kernel.

Coverage is the fixed-truth probability of set membership under the corresponding sampling law.

Szabó, van der Vaart, and van Zanten, §1 and §§2–3

Giné and Nickl, §§6.4 and 8.3

Ray, §4.1