Marginal likelihood and model selection for Gaussian latent tree and forest models

TL;DR

This paper computes the real log-canonical thresholds for Gaussian latent tree and forest models, enabling improved Bayesian model selection by understanding the singularities in their Fisher-information matrices.

Contribution

It introduces a method to compute real log-canonical thresholds for Gaussian latent forest models, connecting algebraic geometry with Bayesian model selection.

Findings

Thresholds can be computed using polyhedral geometry.

Application to model selection improves Bayesian inference.

Demonstrated with simulations and real data.

Abstract

Gaussian latent tree models, or more generally, Gaussian latent forest models have Fisher-information matrices that become singular along interesting submodels, namely, models that correspond to subforests. For these singularities, we compute the real log-canonical thresholds (also known as stochastic complexities or learning coefficients) that quantify the large-sample behavior of the marginal likelihood in Bayesian inference. This provides the information needed for a recently introduced generalization of the Bayesian information criterion. Our mathematical developments treat the general setting of Laplace integrals whose phase functions are sums of squared differences between monomials and constants. We clarify how in this case real log-canonical thresholds can be computed using polyhedral geometry, and we show how to apply the general theory to the Laplace integrals associated with Gaussian latent tree and forest models. In simulations and a data example, we demonstrate how the mathematical knowledge can be applied in model selection.