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Markov chain Monte Carlo (MCMC) is one of the main workhorses ofprobabilistic inference, but it is notoriously hard to measure the quality ofapproximate posterior samples. This challenge is particularly salient in blackbox inference methods, which can hide details and obscure inference failures.In this work, we extend the recently introduced bidirectional Monte Carlotechnique to evaluate MCMC-based posterior inference algorithms. By runningannealed importance sampling (AIS) chains both from prior to posterior and viceversa on simulated data, we upper bound in expectation the symmetrized KLdivergence between the true posterior distribution and the distribution ofapproximate samples. We present Bounding Divergences with REverse Annealing(BREAD), a protocol for validating the relevance of simulated data experimentsto real datasets, and integrate it into two probabilistic programminglanguages: WebPPL and Stan. As an example of how BREAD can be used to guide thedesign of inference algorithms, we apply it to study the effectiveness ofdifferent model representations in both WebPPL and Stan.

Measuring the reliability of MCMC inference with bidirectional Monte Carlo