Bayesian model comparison with un-normalised likelihoods
This addresses a key bottleneck in Bayesian model comparison for fields like computer science and spatial statistics, offering incremental improvements over existing methods.
The paper tackles the challenge of estimating Bayes' factors for models with un-normalized likelihoods, such as Markov random fields, by developing new random weight importance sampling and sequential Monte Carlo methods that circumvent intractable likelihood evaluation, showing in some cases an advantage with biased weight estimates.
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.