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November 2015 On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods
Anne-Marie Lyne, Mark Girolami, Yves Atchadé, Heiko Strathmann, Daniel Simpson
Statist. Sci. 30(4): 443-467 (November 2015). DOI: 10.1214/15-STS523

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.

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Anne-Marie Lyne. Mark Girolami. Yves Atchadé. Heiko Strathmann. Daniel Simpson. "On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods." Statist. Sci. 30 (4) 443 - 467, November 2015. https://doi.org/10.1214/15-STS523

Information

Published: November 2015
First available in Project Euclid: 9 December 2015

zbMATH: 06946197
MathSciNet: MR3432836
Digital Object Identifier: 10.1214/15-STS523

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.30 • No. 4 • November 2015
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