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.

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