An Overview of the Data Augmentation Algorithm

Markov chain Monte Carlo algorithms provide a way of approximately sampling from complicated probability distributions in high dimensions. The data augmentation algorithm is a popular MCMC method which is easy to implement but sometimes suffers from slow convergence. In this report, an overview of the data augmentation algorithm is given, along with a description of two variants that can often result in dramatic improvements in the convergence rates of the underlying Markov chains. A general method based on operator theory is presented to facilitate a theoretical comparison of the convergence rates associated with the above algorithms. The results are illustrated using the Bayesian probit regression model analyzed by Albert and Chib (1993). 1 Background and Motivation for MCMC 1.1 Classical Monte Carlo Methods Suppose we are given a probability distribution π(·) defined on a measurable space (X,B) and we are interested in computing a particular numerical characteristic of π(·), like its mean or standard deviation, but the complexity of the expressions does not allow us to do the computations directly. More precisely, suppose π(·) has density f with respect to a σ-finite measure μ(·) on (X,B). Typically X is an open subset of R and the densities are taken with respect to Lebesgue measure. Suppose we want to estimate expectations of functions g : X→ R with respect to π(·), i.e. we want to estimate π(g) = Eπ[g(x)] = ∫