On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods

TL;DR

This paper explores advanced Bayesian inference techniques for doubly-intractable models, introducing a novel likelihood estimation method using Russian Roulette sampling and pseudo-marginal MCMC, with applications to complex statistical models.

Contribution

It presents a new unbiased likelihood estimation approach based on infinite series and Russian Roulette sampling, applicable to a broad class of doubly-intractable models.

Findings

Demonstrates the method on Ising models and Gaussian Markov Random Fields

Shows the potential of negative likelihood estimates in pseudo-marginal MCMC

Provides a critical assessment of the method's strengths and limitations

Abstract

A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). 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 approximations to the posterior have been suggested, and these are reviewed 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 study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher-Bingham distributions on the $d$-Sphere and a large-scale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.