Index: The Book of Statistical Proofs ▷ General Theorems ▷ Bayesian statistics ▷ Bayesian inference ▷ Prior-averaged coverage of credible sets

Theorem: Let $\theta$ and $X$ have a proper joint Bayesian distribution: $\theta$ has prior probability measure $\Pi$, and $X$ given $\theta$ has sampling law $P_\theta$. Let $0<\alpha<1$. Suppose $C(X)$ is a data-dependent set for which the event $\lbrace \theta\in C(X)\rbrace $ is measurable and

\[\label{eq:cred-calibration-posterior} \begin{gathered} \mathrm{Pr}\{\theta\in C(X)\mid X\}=1-\alpha \\ \text{almost surely under the prior-predictive}\\ \text{law of }X \; . \end{gathered}\]

The prior-predictive law is the marginal distribution of $X$ under the joint Bayesian model. Then, the joint coverage is $1-\alpha$. Equivalently, if $c(\vartheta)=P_\vartheta\lbrace \vartheta\in C(X)\rbrace $ denotes frequentist coverage at the fixed parameter value $\vartheta$,

\[\label{eq:cred-calibration-result} \mathrm{Pr}\{\theta\in C(X)\} =\int c(\vartheta)\,\Pi(\mathrm{d}\vartheta)=1-\alpha \; .\]

The integration variable $\vartheta$ distinguishes a fixed parameter value in a sampling experiment from the random parameter $\theta$ in the Bayesian joint model. The prior-weighted average $\int c(\vartheta)\Pi(\mathrm{d}\vartheta)$ means $\int c(\vartheta)\pi(\vartheta)\,\mathrm{d}\vartheta$ for a prior density $\pi$, or $\sum_\vartheta c(\vartheta)\Pi(\lbrace \vartheta\rbrace )$ for a countable prior. The theorem allows sets of any shape and requires the stated exact posterior probability.

Proof: Define the zero-one indicator $I=\mathbf{1}\lbrace \theta\in C(X)\rbrace $. The expectation of an event’s indicator equals its probability. Conditional on the data, $\mathrm{E}(I\mid X)$ is therefore the posterior probability in \eqref{eq:cred-calibration-posterior}. Total expectation gives

\[\label{eq:cred-calibration-data-first} \mathrm{Pr}\{\theta\in C(X)\} =\mathrm{E}(I) =\mathrm{E}[\mathrm{E}(I\mid X)] =\mathrm{E}(1-\alpha)=1-\alpha \; .\]

Alternatively, apply total expectation while conditioning first on the parameter. Once $\theta=\vartheta$ is fixed, only the data vary, and the expected indicator is $c(\vartheta)$. Thus:

\[\label{eq:cred-calibration-parameter-first} \mathrm{E}(I) =\int \mathrm{E}(I\mid\theta=\vartheta)\,\Pi(\mathrm{d}\vartheta) =\int c(\vartheta)\,\Pi(\mathrm{d}\vartheta) \; .\]

Combining the two ways of calculating the same probability proves \eqref{eq:cred-calibration-result}.

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Sources:
  • George Casella, Roger L. Berger (2002): "Bayesian intervals"; in: Statistical Inference, Second edition, Section 9.2.4, pp. 435-436.

Metadata: ID: P560 | shortcut: cred-calibration | author: kjpetrykowski | date: 2026-10-18, 11:00.