Path Sampling to Compute Integrated Likelihoods: An Adaptive Approach

Computing integrated likelihoods (ILs) to perform Bayesian model selection is a challenging task, particularly when the models considered involve a large number of parameters. In this article, we propose the use of an adaptive quadrature algorithm to automate the selection of the grid in path sampling (PS), an integration technique recognized as one of the most powerful Monte Carlo integration statistical methods for IL estimation. We begin by examining the impact of two tuning parameters of PS, the choice of the auxiliary density and the specification of the grid, both of which are shown to be potentially very influential. We then present the Grid Selection by Adaptive Quadrature (GSAQ) approach and provide a probabilistic bound for the bias of the PS grid estimator in this context. We perform a comparison between the GSAQ and standard grid implementations of PS using two well-studied datasets; GSAQ is found to yield superior results. We then examine other alternatives for PS implementation. In particular, a comparison of the fixed-GSAQ and random approaches to PS is presented. Finally, GSAQ is successfully applied to a longitudinal hierarchical regression model selection problem in multiple sclerosis research. Supplemental materials for this article are available online (see section Supplemental Materials).