Monte Carlo Methods for Complex Problems: From Data Augmentation to Likelihood Free Inference

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Highly complex stochastic models arise frequently in scientific applications. They often lead to statistical inference problems with analytically or computationally intractable likelihood functions. Such problems lie beyond the limit of current Monte Carlo methods. The goal of this proposal is to develop efficient Monte Carlo algorithms for statistical inference problems with intractable likelihoods. Proposed research considers two categories of problems without likelihoods. In the first category, the likelihood is available in analytical forms if the problem is put into an appropriate augmented space. For such problems, a new data augmentation scheme is proposed which leads to a more efficient Markov chain Monte Carlo algorithm. In the second category, the model is a "black box" and only a generating stochastic mechanism is available to simulate data from the model. For such problems, several likelihood-free Monte Carlo algorithms are proposed which extend the power of current Monte Carlo methods. The proposed methods are applied to inference problems in population genetics, panel studies, and hydrological models. The proposed research addresses the urgent need to develop innovative Monte Carlo methodology for problems with intractable likelihood functions. This is of fundamental importance in statistics. It allows researchers to concentrate on scientifically plausible statistical models without worrying about mathematical intractability. Applications of the proposed methods include statistical inference in molecular population genetics which can help locating genes that are responsible for genetic diseases, and Bayesian calibration of hydrological models which can be used to predict ground-water flow. The proposed research has significant impact on education through involvement of graduate and undergraduate students directly in the proposed research, incorporation of proposed algorithms into related courses, and dissemination of research results to the scientific communities.