Bayesian information criteria for clustering normally distributed data

TL;DR

This paper introduces an exact Bayesian information criterion for clustering normally distributed data, addressing limitations of traditional BIC in small sample clusters, and providing a more reliable statistical tool for cluster determination.

Contribution

It develops an exact Bayesian information criterion for clustering, avoiding approximations and improving cluster analysis accuracy over traditional BIC methods.

Findings

Exact BIC derived for clustering without Laplace approximation

Traditional BIC unsuitable for small clusters, new criterion addresses this

Provides a more reliable statistical method for determining the number of clusters

Abstract

Maximum likelihood estimates (MLEs) are asymptotically normally distributed, and this property is used in meta-analyses to test the heterogeneity of estimates, either for a single cluster or for several sub-groups. More recently, MLEs for associations between risk factors and diseases have been hierarchically clustered to search for diseases with shared underlying causes, but an objective statistical criterion is needed to determine the number and composition of clusters. To tackle this problem, conventional statistical tests are briefly reviewed, before considering the posterior distribution for a partition of data into clusters. The posterior distribution is calculated by marginalising out the unknown cluster centres, and is different to the likelihood associated with mixture models. The calculation is equivalent to that used to obtain the Bayesian Information Criterion (BIC), but is exact, without a Laplace approximation. The result includes a sum of squares term, and terms that depend on the number and composition of clusters, that penalise the number of free parameters in the model. The usual BIC is shown to be unsuitable for clustering applications unless the number of items in each individual cluster is sufficiently large.