Marginal likelihood computation for model selection and hypothesis testing: an extensive review

TL;DR

This paper provides a comprehensive review of methods for computing marginal likelihoods, essential for model selection and hypothesis testing across various scientific fields, highlighting their advantages, limitations, and recent developments.

Contribution

It offers an extensive overview of current techniques for marginal likelihood computation, including theoretical insights, comparisons, and discussions on handling improper priors.

Findings

Comparison of key methodologies through numerical experiments

Identification of limitations and benefits of different techniques

Discussion on issues related to improper priors

Abstract

This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.