Understanding and improving Markov chain Monte Carlo methods in high dimensional problems

Complicated probability distributions arise routinely in realistic statistical modeling, accounting for complex data types, missing data, and high dimensionality of parameters of interest. Markov chain Monte Carlo (MCMC) methods have been widely used to estimate intractable integrals over these distributions, which comprises a vital step for performing statistical inference and prediction. Unfortunately, efficient and reliable MCMC methods are currently lacking for problems involving high dimensional probability distributions, which have increasingly emerged in many areas like genetics, image processing, and ecology. This thesis focuses on understanding and improving MCMC methods in high dimensional problems.

We first develop statistically and computationally efficient two-block Gibbs samplers for three commonly used Bayesian shrinkage models. A blocking technique is used in this procedure to deal with the high dimensionality of the corresponding posterior distributions. We also conduct a theoretical investigation on standard error evaluations of Monte Carlo estimates based on Markov chain samples. In particular, we directly obtain Markov chain central limit theorems based on convergence rates in Wasserstein distance, which enriches the toolbox for analyzing MCMC methods.