  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.
