Diagnostics for Monte Carlo Algorithms for Models with Intractable Normalising Functions

Abstract

Models with intractable normalizing functions have numerous applications ranging from network models to image analysis to spatial point processes. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as the asymptotic distribution. Other "asymptotically inexact" algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms, and hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, applies in principle to any asymptotically exact or inexact algorithm. We develop an approximate version of this new diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods. We apply our diagnostics to several algorithms in the context of challenging simulated and real data examples, including an Ising model, an exponential random graph model, and a Markov point process.