## Bernoulli

- Bernoulli
- Volume 23, Number 2 (2017), 1202-1232.

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

Mathias Drton, Shaowei Lin, Luca Weihs, and Piotr Zwiernik

#### 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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 2 (2017), 1202-1232.

**Dates**

Received: December 2014

Revised: September 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1486177397

**Digital Object Identifier**

doi:10.3150/15-BEJ775

**Mathematical Reviews number (MathSciNet)**

MR3606764

**Zentralblatt MATH identifier**

1381.62224

**Keywords**

algebraic statistics Gaussian graphical model latent tree models marginal likelihood multivariate normal distribution singular learning theory

#### Citation

Drton, Mathias; Lin, Shaowei; Weihs, Luca; Zwiernik, Piotr. Marginal likelihood and model selection for Gaussian latent tree and forest models. Bernoulli 23 (2017), no. 2, 1202--1232. doi:10.3150/15-BEJ775. https://projecteuclid.org/euclid.bj/1486177397