Bayesian methods are now routinely used in very complex models, with posterior distributions estimated by Markov chain Monte Carlo (MCMC) methods. There are two consequences to this. First, complex Bayesian models are virtually always governed by some hyperparameters, which have a large impact on subsequent inference. Therefore, there is now a strong need for methods that enable selection of these hyperparameters. Second, the Markov chains used to estimate the posterior distributions now run in non-standard spaces, for example large function spaces, and there is a need for the development of MCMC methods that will work well in non-standard spaces. The investigators develop methods for efficiently estimating marginal likelihoods for large number of hyperparameter values. This will enable implementation of the empirical Bayes method, and also enables users to determine classes of hyperparameter values which constitute reasonable choices. The exploration of intractable posterior distributions resulting from complex Bayesian models often requires MCMC. Unfortunately, in contrast with classical Monte Carlo, establishing central limit theorems (CLTs) for MCMC estimators is not straightforward. This is a serious practical problem because the ability to choose an appropriate MCMC sample size hinges upon the existence of a CLT. The investigators use spectral methods to develop checkable sufficient conditions for CLTs as well as methods for comparing the asymptotic efficiency of MCMC algorithms with the same target distribution. They apply the theoretical results to very concrete problems of model selection and assessment.
Model selection in complex situations is an important and pervasive problem in scientific and medical research. It includes in particular variable selection in regression, where a few important variables are to be selected from many candidates and used for understanding, prediction and decision making. Different models can lead to different conclusions, with potential impact on public policy. The investigators develop efficient computational methods for determining optimal models in complex settings. The project has an educational component in that graduate students are involved in the research under the supervision of the investigators.