Practical issues in modern Monte Carlo integration

Computing marginal likelihoods to perform Bayesian model selection is a challenging task, particularly when the models considered involve a large number of parameters. In this thesis, we propose the use of an adaptive quadrature algorithm to automate the selection of the grid in path sampling, an integration technique recognized as one of the most powerful Monte Carlo integration statistical methods for marginal likelihood estimation. We begin by examining the impact of two tuning parameters of path sampling, the choice of the importance density and the specification of the grid, which are both shown to be potentially very influential. We then present, in detail, the Grid Selection by Adaptive Quadrature (GSAQ) algorithm for selecting the grid. We perform a comparison between the GSAQ and standard grid implementation of path sampling using two well-studied data sets; the GSAQ approach is found to yield superior results. GSAQ is then successfully applied to a longitudinal hierarchical regression model selection problem in Multiple Sclerosis research.Using an identity arising in path sampling, we then derive general expressions for the Kullback-Leibler (KL) and Jeffrey (J) divergences between two distributions with common support but from possibly different parametric families. These expressions naturally stem from path sampling when the popular geometric path is used to link the extreme densities. Expressions for the KL and J-divergences are also given for any two intermediate densities lying on the path. Estimates for the KL divergence (up to a constant) and for the J-divergence, between a posterior distribution and a selected importance density, can be obtained directly, prior to path sampling implementation. The J-divergence is shown to be helpful for choosing importance densities that minimize the error of the path sampling estimates.Finally we present the results of a simulation study devised to investigate whether improvement in performance can be achieved by using the KL and J-divergences to select sequences of distributions in parallel (population-based) simulations, such as in the Sequential Monte Carlo Sampling and the Annealed Importance Sampling algorithms. We compare these choices of sequences to more conventional choices in the context of a mixture example. Unexpected results are obtained, and those for the KL and J-divergences are mixed. More fundamentally, we uncover the need to select the sequence of tempered distributions in accordance with the resampling scheme