A Short Note on the Efficiency of Markov Chains for Bayesian Linear Regression Models with Heavy-Tailed Errors

TL;DR

This paper investigates the efficiency of Markov chain Monte Carlo methods for Bayesian linear regression models with heavy-tailed errors, demonstrating geometric ergodicity under various conditions.

Contribution

It extends existing results by proving geometric ergodicity of samplers in heavy-tailed error models and improper prior cases, with uniform ergodicity established.

Findings

Sampler remains geometrically ergodic with heavy-tailed errors.

Uniform ergodicity is verified via minorization condition.

Milder conditions ensure ergodicity in improper prior cases.

Abstract

In this short note, we consider posterior simulation for a linear regression model when the error distribution is given by a scale mixture of multivariate normals. We first show that the sampler of Backlund and Hobert (2020) for the case of the conditionally conjugate normal-inverse Wishart prior continues to be geometrically ergodic even when the error density is heavier-tailed. Moreover, we prove that the ergodicity is uniform by verifying the minorization condition. In the second half of this note, we treat an improper case and show that the sampler of Section 4 of Roy and Hobert (2010) is geometrically ergodic under significantly milder conditions.