Index: The Book of Statistical Proofs ▷ General Theorems ▷ Estimation theory ▷ Interval estimates ▷ Prior predictive calibration of Bayesian 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

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

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

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

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:posterior}. Applying the law of total expectation gives

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

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:

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

Combining \eqref{eq:data-first} and \eqref{eq:parameter-first}, this proves \eqref{eq: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: crs-priorpred | author: kjpetrykowski | date: 2026-10-18, 11:00.