Robust Bayesian Graphical Modeling Using Dirichlet t-Distributions

Abstract

Bayesian graphical modeling provides an appealing way to obtain uncertainty estimates when inferring network structures, and much recent progress has been made for Gaussian models. For more robust inferences, it is natural to consider extensions to $t$-distribution models. We argue that the classical multivariate $t$-distribution, defined using a single latent Gamma random variable to rescale a Gaussian random vector, is of little use in more highly multivariate settings, and propose other, more flexible $t$-distributions. Using an independent Gamma-divisor for each component of the random vector defines what we term the alternative $t$-distribution. The associated model allows one to extract information from highly multivariate data even when most experiments contain outliers for some of their measurements. However, the use of this alternative model comes at increased computational cost and imposes constraints on the achievable correlation structures, raising the need for a compromise between the classical and alternative models. To this end we propose the use of Dirichlet processes for adaptive clustering of the latent Gamma-scalars, each of which may then divide a group of latent Gaussian variables. The resulting Dirichlet $t$-distribution interpolates naturally between the two extreme cases of the classical and alternative $t$-distributions and combines more appealing modeling of the multivariate dependence structure with favorable computational properties.

This paper was invited by the Editor-in-Chief of Bayesian Analysis to be presented as the 2014 Best Bayesian Analysis Paper at the Twelfth World Meeting of the International Society for Bayesian Analysis (ISBA2014), held in Cancun, Mexico, on July 14–18, 2014, with invited discussions by Babak Shahbaba and François Caron.