A BGe score for tied-covariance mixtures of Gaussian Bayesian networks

TL;DR

This paper introduces a new BGe score for tied-covariance Gaussian Bayesian network mixtures, deriving an analytic marginal likelihood and implementing MCMC inference, with empirical comparisons to full-covariance models.

Contribution

It proposes a tied-covariance mixture model for Gaussian Bayesian networks and derives its analytic marginal likelihood, enabling new scoring and inference methods.

Findings

The tied-covariance BGe score is analytically derived.

MCMC inference effectively combines structure and mixture sampling.

Empirical results compare tied- and full-covariance models on data.

Abstract

Mixtures of Gaussian Bayesian networks have previously been studied under full-covariance assumptions, where each mixture component has its own covariance matrix. We propose a mixture model with tied-covariance, in which all components share a common covariance matrix. Our main contribution is the derivation of its marginal likelihood, which remains analytic. Unlike in the full-covariance case, however, the marginal likelihood no longer factorizes into component-specific terms. We refer to the new likelihood as the BGe scoring metric for tied-covariance mixtures of Gaussian Bayesian networks. For model inference, we implement MCMC schemes combining structure MCMC with a fast Gibbs sampler for mixtures, and we empirically compare the tied- and full-covariance mixtures of Gaussian Bayesian networks on simulated and benchmark data.