Grant: Discovery Grants Program - Individual

Markov chain Monte Carlo (MCMC) algorithms are widely used in the scientific community. They are used to generate approximate samples from complicated target distributions resulting from statistical and applied probability models. In particular, generic algorithms such as Metropolis and Langevin algorithms are very popular because of the ease of implementations. However, it is well known that choosing the best proposal distribution is not an easy task. Failing to do that can result in an inefficient algorithm and misleading results. My research program will focus on the theory and implementation of MCMC algorithms, typically on unbounded Euclidean state spaces, motivated by distributions such as those encountered in Bayesian inference. The three main objectives of my proposed research program are: 1. Bounding the convergence rates quantitatively. I shall generalize and combine available techniques through the application of decomposition theorems, from discrete state spaces, to general state spaces. This work will have direct value for theoreticians working in MCMC and will hopefully unify approaches for estimating and optimizing rates of convergence of algorithms; 2. Optimizing the convergence rates by proper scaling of the proposal distribution, when the dimension of the target density approaches infinity. I shall continue to focus on local MCMC algorithms, loosely referring to methods which have increment distributions centered at (or close to) the current state, and on proving weak convergence results for a wider class of target and proposal distributions. This work will extend and refine the scaling guidelines from available theory for all practitioners in the area; 3. Suggest modifications and study practical implementation issues on some existing sampling algorithms (e.g. MCMC, adaptive MCMC, etc.) widely used in the scientific literature. This includes stabilities of adaptations, convergence and efficiencies of these algorithms, and other problems motivated by practitioners. This work, which is foundational, will enrich the theories of MCMC methods and related extensions.