Joint $p$-Values for Higher-Powered Bayesian Model Checking with Frequentist Guarantees

TL;DR

This paper proposes a joint posterior p-value method for Bayesian model checking that maintains frequentist guarantees and improves power in high-dimensional settings by addressing conservativeness of traditional p-values.

Contribution

It introduces a joint posterior p-value that extends existing methods, overcoming conservativeness issues in high dimensions with negative association of test statistics.

Findings

Joint p-value improves power over traditional methods.

Method maintains frequentist guarantees.

Validated through simulation examples.

Abstract

We introduce a joint posterior $p$-value, an extension of the posterior predictive $p$-value for multiple test statistics, designed to address limitations of existing Bayesian $p$-values in the setting of continuous model expansion. In particular, we show that the posterior predictive $p$-value, as well as its sampled variant, become more conservative as the parameter dimension grows, and we demonstrate the ability of the joint $p$-value to overcome this problem in cases where we can select test statistics that are negatively associated under the posterior. We validate these conclusions with a pair of simulation examples in which the joint $p$-value achieves substantial gains to power with only a modest increase in computational cost.