A Tutorial on Bridge Sampling

TL;DR

This tutorial explains bridge sampling, a reliable numerical method for estimating marginal likelihoods in Bayesian models, demonstrating its accuracy and applicability to complex models in psychology.

Contribution

It provides a clear tutorial on bridge sampling, illustrating its use with examples and applying it to real models, highlighting its effectiveness for high-dimensional Bayesian model comparison.

Findings

Bridge sampling yields accurate marginal likelihood estimates.

It is effective for both simple and hierarchical models.

The method is accessible for researchers in psychology.

Abstract

The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model---a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.