A Bayesian information criterion for singular models

TL;DR

This paper introduces a practical extension of the Bayesian information criterion (BIC) tailored for singular models, which are common in mixture models and latent factor analysis, addressing limitations of traditional BIC.

Contribution

It proposes a new BIC extension specifically designed for singular models, resolving theoretical paradoxes and improving model selection accuracy in complex statistical models.

Findings

The new criterion performs better in selecting the correct number of components in mixture models.

It provides a consistent model selection method for singular models.

The approach is applicable to latent factor and reduced-rank regression models.

Abstract

We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher-information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion (BIC) and the penalty structure in BIC generally does not reflect the frequentist large-sample behavior of their marginal likelihood. While large-sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components of mixture models, the number of factors in latent factor models or the rank in reduced-rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems.