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In large-scale genomic applications vast numbers of molecular features arescanned in order to find a small number of candidates which are linked to aparticular disease or phenotype. This is a variable selection problem in the"large p, small n" paradigm where many more variables than samples areavailable. Additionally, a complex dependence structure is often observed amongthe markers/genes due to their joint involvement in biological processes andpathways. Bayesian variable selection methods that introduce sparseness throughadditional priors on the model size are well suited to the problem. However,the model space is very large and standard Markov chain Monte Carlo (MCMC)algorithms such as a Gibbs sampler sweeping over all p variables in eachiteration are often computationally infeasible. We propose to employ thedependence structure in the data to decide which variables should always beupdated together and which are nearly conditionally independent and hence donot need to be considered together. Here, we focus on binary classificationapplications. We follow the implementation of the Bayesian probit regressionmodel by Albert and Chib (1993) and the Bayesian logistic regression model byHolmes and Held (2006) which both lead to marginal Gaussian distributions. Wein- vestigate several MCMC samplers using the dependence structure in differentways. The mixing and convergence performances of the resulting Markov chainsare evaluated and compared to standard samplers in two simulation studies andin an application to a real gene expression data set.

MCMC algorithms for Bayesian variable selection in the logistic regression model for large-scale genomic applications