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We introduce a joint posterior $p$-value, an extension of the posteriorpredictive $p$-value for multiple test statistics, designed to addresslimitations of existing Bayesian $p$-values in the setting of continuous modelexpansion. In particular, we show that the posterior predictive $p$-value, aswell as its sampled variant, become more conservative as the parameterdimension grows, and we demonstrate the ability of the joint $p$-value toovercome this problem in cases where we can select test statistics that arenegatively associated under the posterior. We validate these conclusions with apair of simulation examples in which the joint $p$-value achieves substantialgains to power with only a modest increase in computational cost.

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