  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.
