Large Sample Analysis of Markov Chain Monte Carlo Methods in Bayesian Statistics From a Frequentist Perspective

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This project concerns Markov Chain Monte Carlo (MCMC), which is a class of computer algorithms that are widely used to simulate complicated probability distributions. In the field of statistics, probability distributions are often used to model uncertainty about unknown parameters that one wishes to estimate, for example, the average household income of a large nation, or the difference in average life expectancy between two demographic groups. The estimation procedure is based on a data set, which is usually a sample collected from an underlying population through a survey or experiment. MCMC is widely regarded as an extremely powerful tool for high-quality estimation, but not enough is known about its reliability when the sample is massive. This project aims to study the mathematical properties of various MCMC algorithms in large sample settings. Results of the research are expected to advance understanding in the performance of MCMC algorithms that are applied to modern data sets in fields such as economics, biology, and astronomy. Undergraduate and graduate students involved in the research efforts of this project will receive training in probability theory as well as mathematical and applied statistics. More specifically, this project focuses on MCMC algorithms that are used to explore posterior distributions in Bayesian statistical models. From a frequentist perspective, the data set associated with a Bayesian model is assumed to be generated from an underlying distribution. The Markov transition kernel of an MCMC algorithm can thus be regarded as a statistic, i.e., observable random element, no different from a classical vector-valued statistic, e.g., a sample mean. Just like with any statistic, it is important to understand the asymptotic behavior of an MCMC transition kernel when the sample size of the data set grows. This project aims to develop general theories for the problem using techniques from classical large sample theory in conjunction with those from Markov chain theory. Special attention will be given to data augmentation algorithms, which are a broad class of practically relevant MCMC algorithms that exhibit rich and meaningful large sample properties. The project will also involve studying the mixing times of MCMC algorithms in the large sample regime, which is crucial to the effectiveness of MCMC-based inference in practice. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.