We discuss novel approaches to evaluation of both upper and lower bounds on log marginal likelihoods for model comparison in Bayesian analysis. From posterior Monte Carlo samples, we show how existing variational approximation methods defining lower bounds on marginal likelihoods can be extended to also define upper bounds, and develop optimization methods to minimize such upper bounds. Further, using this new approach to upper bound evaluation, we suggest and exemplify a new quasi-optimized lower bound that can often be obtained with trivial computations compared to current methods. We further discuss the use of partial analytic marginalization of some model parameters as a way of significantly reducing the differences between upper and lower bounds to improve marginal likelihood approximation. To implement this, however, traditional variational methods are intractable, and we provide solution in terms of a novel Monte Carlo Stochastic Approximation (MCSA). We provide theoretical results on convergence of the resulting approximations to true bounds, and several simulation examples in regression and mixture models to demonstrate the accuracy and efficacy of the new methods.

In cancer studies as in many areas of human disease research, gene expression microarray technology has been central to the emergent field of genomic medicine. Expression profiles of physiological states and clinical outcomes play increasing roles as biomarkers in both experimental and human observational studies. Central challenges in moving towards clinical applications include hard questions of how to link and combine such measures across contexts: from laboratory experiments with cultured cells, to animal model experiments, to human outcome studies and clinical trials. The question of how to translate and transfer experimental, laboratory findings to the context of human observational studies sits at the core of current translational research agendas. This case study focuses on precisely this question in cancer genomics, where the in vitro laboratory results involve gene expression signatures of changes in human cells in response to a set of interventions on cancer related genes, the oncogene intervention experiments, and the in vivo context is gene expression studies with data generated from human breast tumours. The analyses involve a series of applications of sparse Bayesian latent factor regression models, and are illustrative of the use of these models for large-scale multivariate data arising from both