Submitted to the Statistical Science On Russian Roulette Estimates for Bayesian inference with Doubly-Intractable Likelihoods∗

Abstract. A large number of statistical models are ‘doubly-intractable’: the likelihood normalising term, which is a function of the model pa-rameters, is intractable, as well as the marginal likelihood (model evi-dence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic ap-proximations to the posterior have been suggested, and these are re-viewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors. In addition, taking inspiration from the Physics literature, we stud