Bayesian model comparison with un-normalised likelihoods

TL;DR

This paper introduces new Monte Carlo methods for Bayesian model comparison in models with intractable likelihoods, such as Markov random fields, by using simulation-based estimates of Bayes' factors, including biased weights.

Contribution

It proposes novel importance sampling and sequential Monte Carlo techniques for estimating Bayes' factors without evaluating intractable likelihoods, and compares their performance to existing methods.

Findings

Biased weight estimates can sometimes improve efficiency.

The new methods perform well in certain intractable likelihood models.

Caution is advised when using biased estimates due to potential biases.

Abstract

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalising constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes' factors (BFs) for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.