On the Data Augmentation Algorithm for Bayesian Multivariate Linear Regression with Non-Gaussian Errors

TL;DR

This paper extends the analysis of a data augmentation algorithm for Bayesian multivariate linear regression with non-Gaussian errors, providing convergence rate results for the Markov chain in the multivariate setting.

Contribution

It generalizes previous univariate convergence results to the multivariate case, offering conditions for geometric ergodicity of the Markov chain.

Findings

Provides convergence rate analysis for the multivariate case

Establishes sufficient conditions for geometric ergodicity

Extends prior univariate results to multivariate models

Abstract

Let $\pi$ denote the intractable posterior density that results when the likelihood from a multivariate linear regression model with errors from a scale mixture of normals is combined with the standard non-informative prior. There is a simple data augmentation algorithm (based on latent data from the mixing density) that can be used to explore $\pi$. Hobert et al. (2015) [arXiv:1506.03113v1] recently performed a convergence rate analysis of the Markov chain underlying this MCMC algorithm in the special case where the regression model is univariate. These authors provide simple sufficient conditions (on the mixing density) for geometric ergodicity of the Markov chain. In this note, we extend Hobert et al.'s (2015) result to the multivariate case.