Bayesian Semiparametric Multivariate Density Regression with Coordinate-Wise Predictor Selection

TL;DR

This paper introduces a Bayesian density regression method using Gaussian copulas and tensor factorization to model multivariate outcomes with covariate selection, scalable to many covariates.

Contribution

It develops a novel Tucker tensor factorization-based structure for covariate effects and a flexible covariate aggregation mechanism, enhancing scalability and interpretability.

Findings

Effective in simulation studies

Applied successfully to NHANES dietary data

Reduces computational complexity with many covariates

Abstract

We propose a flexible Bayesian approach for estimating the joint density of a multivariate outcome of interest in the presence of categorical covariates. Leveraging a Gaussian copula framework, our method effectively captures the dependence structure across different coordinates of the multivariate response. The conditional (on covariates) marginal (across outcomes) distributions are modeled as flexible mixtures with shared atoms across coordinates, while the mixture weights are allowed to vary with covariates through a novel Tucker tensor factorization-based structure, which enables the identification of coordinate-specific subsets of influential covariates. In particular, we replace the traditional mode matrices with coordinate-specific random partition models on the covariate levels, offering a flexible mechanism to aggregate covariate levels that exhibit similar effects on the response. Additionally, to handle settings with many covariates, we introduce a Markov chain Monte Carlo algorithm that scales with the number of aggregated levels rather than the original levels, significantly reducing memory requirements and improving computational efficiency. We demonstrate the method's numerical performance through simulation experiments and its practical applicability through the analysis of NHANES dietary data.