Diagnostics for Monte Carlo Algorithms for Models with Intractable Normalising Functions

Models with intractable normalising functions have numerous applications. Because the normalising constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. Many 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. We propose two new diagnostics that address these problems. We provide theoretical justification for our methods and apply them to several algorithms in the context of challenging examples.