Convergence Analysis of Data Augmentation Algorithms for Bayesian Robust Multivariate Linear Regression with Incomplete Data

TL;DR

This paper analyzes the convergence properties of data augmentation algorithms in Bayesian robust multivariate linear regression with incomplete data, providing conditions for ergodicity and insights on imputation strategies.

Contribution

It offers new theoretical results on the convergence of data augmentation algorithms under different data missingness structures in Bayesian regression.

Findings

Conditions for geometric ergodicity with monotone missing data

Sufficient conditions for Harris ergodicity without monotone structure

Intermediate imputation can slow convergence, while post hoc imputation does not

Abstract

Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a "monotone" structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided.