The Art Of Data Augmentation And Parameter Expansion In Markov Chain Monte Carlo

Abstract

Markov Chain Monte Carlo (MCMC) method plays a crucial role in Bayesian inference but suffers inefficiencies in high-dimensional scenarios. In this report, we summarize recent developments in integrating Data Augmentation (DA) and Parameter Expansion (PE) techniques to enhance MCMC efficiency. By leveraging left-(invariant) Haar measures on locally compact groups, we provide a precise definition of the Parameter Expansion Data Augmentation (PX-DA) algorithm. This novel approach refines the traditional DA methods and exhibits improved convergence properties, as supported by theoretical analysis and extensive simulations, and contributes to advancing Bayesian methods, providing a more robust framework for handling complex models.