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Simulation-based calibration checking (SBC) is a practical method to validatecomputationally-derived posterior distributions or their approximations. Inthis paper, we introduce a new variant of SBC to alleviate several knownproblems. Our variant allows the user to in principle detect any possible issuewith the posterior, while previously reported implementations could neverdetect large classes of problems including when the posterior is equal to theprior. This is made possible by including additional data-dependent testquantities when running SBC. We argue and demonstrate that the joint likelihoodof the data is an especially useful test quantity. Some other types of testquantities and their theoretical and practical benefits are also investigated.We provide theoretical analysis of SBC, thereby providing a more completeunderstanding of the underlying statistical mechanisms. We also bring attentionto a relatively common mistake in the literature and clarify the differencebetween SBC and checks based on the data-averaged posterior. We support ourrecommendations with numerical case studies on a multivariate normal exampleand a case study in implementing an ordered simplex data type for use withHamiltonian Monte Carlo. The SBC variant introduced in this paper isimplemented in the $\mathtt{SBC}$ R package.

Simulation-Based Calibration Checking for Bayesian Computation: The Choice of Test Quantities Shapes Sensitivity