Bayesian Tensor Decompositions and Sparse Log-Linear Models

• Activity Number: 381
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): James Johndrow*+ and Anirban Bhattacharya and David Dunson
• Companies: and Texas A&M and Duke University
• Keywords: Bayesian ; Categorical data ; log-linear model ; tensor decomposition ; contingency table ; sparsity

Abstract

Bayesian analysis of contingency tables routinely proceeds by specifying a prior on the parameters of a log-linear model, with latent structure models providing a common alternative. Latent structure models induce a low rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between notions of dimensionality reduction in the two paradigms. We derive several results relating the support of a log-linear model to the nonnegative rank of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of Bayesian latent structure models, which bridge existing PARAFAC and Tucker decomposition priors, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. We propose a Gibbs sampling algorithm for posterior computation, and illustrate advantages of the new prior in simulations and an application to functional disability data