Sampling from Distributions with Intractable Integrals

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

During the past five decades, Markov chain Monte Carlo (MCMC) methods have been developed as a versatile and powerful tool for scientific computing. However, as known by many researchers, conventional MCMC methods suffer from the inability to sample from distributions with intractable integrals. The goal of this project is to develop some innovative Monte Carlo algorithms which are capable of sampling from distributions with intractable integrals. To achieve this goal, the PI proposes a new population Monte Carlo algorithm---Monte Carlo dynamically weighted importance sampling (MCDWIS). In simulations, MCDWIS replaces the ratio of intractable integrals by its Monte Carlo estimate, and the bias introduced thereby is counterbalanced by giving different weights to new samples produced. MCDWIS allows for the use of Monte Carlo estimates in MCMC simulations, while leaving the target distribution invariant with respect to important weights. Unlike auxiliary variable MCMC methods, MCDWIS avoids the requirement for perfect samples, and thus can be applied to many statistical models for which perfect sampling is unavailable or very expensive. As discussed in the proposal, MCDWIS can also be used to sample from incomplete posterior distributions for missing data and random effects-related models (e.g., generalized linear mixed models), which are traditionally treated with the expectation-maximization (EM) or Monte Carlo EM algorithms. In addition to providing a fully Bayesian analysis for these models, the MCDWIS can potentially overcome, due to its self-adjusting mechanism, the local-trap problem suffered by the EM and Monte Carlo EM algorithms. In this proposal, the PI also proposes an importance sampling-targeted stochastic approximation Monte Carlo algorithm, the so-called importance stochastic approximation Monte Carlo algorithm, which can be used for Bayesian inference for the models with intractable normalizing constants. The intellectual merit of this project is to provide some innovative computational methods, which are expected to play a major role in statistical inference for an important class of scientific models, including random graph models used in social network analysis, autonormal models used in spatial data analysis, autologistic models used in disease mapping, and generalized linear mixed models used in biomedical data analysis, among others. Successful inferences of the models will enhance people's underderstanding to the underlying natural, social, or biological systems. This project will have broader impacts in both communities of statistical methodology and scientific computing. The research results will be disseminated to these communities via direct collaboration with researchers in other disciplines, conference presentations, books, and papers to be published in academic journals. The project will have also significant impacts on education through direct involvement of graduate students in the project and incorporation of results into undergraduate and graduate courses.