Bayes-optimal prediction with frequentist coverage control

TL;DR

This paper presents a method for constructing prediction regions that optimally incorporate prior information to achieve exact frequentist coverage, applicable to a wide range of models without relying on asymptotics.

Contribution

It introduces a framework for Bayes-optimal prediction regions that maintain coverage regardless of prior accuracy, applicable to models with complete sufficient statistics.

Findings

Prediction regions with lower volume when prior info is accurate

Coverage is maintained even if prior info is inaccurate

Applicable to parametric and nonparametric models, including regression

Abstract

This article illustrates how indirect or prior information can be optimally used to construct a prediction region that maintains a target frequentist coverage rate. If the indirect information is accurate, the volume of the prediction region is lower on average than that of other regions with the same coverage rate. Even if the indirect information is inaccurate, the resulting region still maintains the target coverage rate. Such a prediction region can be constructed for models that have a complete sufficient statistic, which includes many widely-used parametric and nonparametric models. Particular examples include a Bayes-optimal conformal prediction procedure that maintains a constant coverage rate across distributions in a nonparametric model, as well as a prediction procedure for the normal linear regression model that can utilize a regularizing prior distribution, yet maintain a frequentist coverage rate that is constant as a function of the model parameters and explanatory variables. No results in this article rely on asymptotic approximations.