Existence of matching priors on compact spaces

TL;DR

This paper investigates the conditions under which matching priors exist on compact spaces, providing topological criteria and methods to construct approximate matching priors for common credible regions.

Contribution

It establishes topological conditions for the existence of matching priors on compact spaces and proposes a numerical scheme for their approximation.

Findings

Matching priors exist if the rejection-probability function is continuous under the Wasserstein metric.

Common credible regions like credible balls often do not meet the topological conditions.

A numerical scheme for approximating matching priors is evaluated.

Abstract

A matching prior at level $1-\alpha$ is a prior such that an associated $1-\alpha$ credible set is also a $1-\alpha$ confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives topological conditions under which matching priors for specific families of credible regions exist. Informally, we prove that, on compact parameter spaces, a matching prior exists if the so-called rejection-probability function is jointly continuous when we adopt the Wasserstein metric on priors. In light of this general result, we observe that typical families of credible regions, such as credible balls, highest-posterior density regions, quantiles, etc., fail to meet this topological condition. We show how to design approximate posterior credible balls and highest-posterior-density regions that meet these topological conditions, yielding matching priors. Finally, we evaluate a numerical scheme for computing approximately matching priors based on discretization and iteration. The proof of our main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric that may be of independent interest.