Investigation of the widely applicable Bayesian information criterion

TL;DR

This paper evaluates the practical performance of the widely applicable Bayesian information criterion (WBIC), highlighting its strengths with informative priors and its tendency to overestimate evidence in small samples across various models.

Contribution

The study provides an empirical assessment of WBIC's effectiveness across different models, including singular models, and compares it to thermodynamic integration.

Findings

WBIC performs well with informative priors.

It tends to overestimate evidence in small samples.

Performance varies between regular and singular models.

Abstract

The widely applicable Bayesian information criterion (WBIC) is a simple and fast approximation to the model evidence that has received little practical consideration. WBIC uses the fact that the log evidence can be written as an expectation, with respect to a powered posterior proportional to the likelihood raised to a power $t^*\in{(0,1)}$, of the log deviance. Finding this temperature value $t^*$ is generally an intractable problem. We find that for a particular tractable statistical model that the mean squared error of an optimally-tuned version of WBIC with correct temperature $t^*$ is lower than an optimally-tuned version of thermodynamic integration (power posteriors). However in practice WBIC uses the a canonical choice of $t=1/\log(n)$. Here we investigate the performance of WBIC in practice, for a range of statistical models, both regular models and singular models such as latent variable models or those with a hierarchical structure for which BIC cannot provide an adequate solution. Our findings are that, generally WBIC performs adequately when one uses informative priors, but it can systematically overestimate the evidence, particularly for small sample sizes.