PhD student, University of Minnesota, School of Statistics
Convergence Analysis of Data Augmentation Algorithms for Bayesian Robust Multivariate Linear Regression with Incomplete Data
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.
Haoxiang Li is a fourth year Ph.D. student in statistics at the University of Minnesota (UMN), advised by Prof. Qian Qin and Prof. Galin Jones. Prior to UMN, He received his bachelor’s degree in mathematics from Beijing Normal University. His main research interests are Bayesian statistics and Markov chain Monte Carlo (MCMC). In particular, he works on convergence analysis and reliability assessment tools for MCMC algorithms.