On predictive probability matching priors

TL;DR

This paper explores priors that align Bayesian and frequentist predictive probabilities, especially focusing on uniformly predictive matching priors (UPMPs) in multi-parameter models, analyzing their form, existence, and uniqueness.

Contribution

It extends the analysis of UPMPs from scalar to multi-parameter cases, providing new results on their structure and conditions for existence and uniqueness.

Findings

UPMPs can depend on the level α in multi-parameter models.

In scalar cases, Jeffreys' prior is the unique UPMP for quantile matching.

The paper characterizes the form of UPMPs for both quantile and density matching in multi-parameter settings.

Abstract

We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply as devices for producing frequentist prediction regions. Here we analyse the $O(n^{-1})$ term in the expansion of the coverage probability of a Bayesian prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike the situation for parametric matching, asymptotic predictive matching priors may depend on the level $\alpha$. We investigate uniformly predictive matching priors (UPMPs); that is, priors for which this $O(n^{-1})$ term is zero for all $\alpha$. It was shown in [Ann. Statist. 28 (2000) 1414--1426] that, in the case of quantile matching and a scalar parameter, if such a prior exists then it must be Jeffreys' prior. In the present article we investigate UPMPs in the multiparameter case and present some general results about the form, and uniqueness or otherwise, of UPMPs for both quantile and highest predictive density matching.