Alternative prior distributions for variable selection with very many more variables than observations

The problem of variable selection in regression and the generalised linear model is addressed. We adopt a Bayesian approach with priors for the regression coefficients that are scale mixtures of normal distributions and embody a high prior probability of proximity to zero. By seeking modal estimates we generalise the lasso. Properties of the priors and their resultant posteriors are explored in the context of the linear and generalised linear model especially when there are more variables than observations. We develop EM algorithms that embrace the need to explore the multiple modes of the non log-concave posterior distributions. Finally we apply the technique to microarray data using a probit model to find the genetic predictors of osteo- versus rheumatoid arthritis. Keywords: Bayesian modal analysis, Variable selection in regression, Scale mixtures of normals, Improper Jeffreys prior, lasso, Penalised likelihood, EMalgorithm, Multiple modes, More variables than observations, Singular value decomposition, Latent variables, Probit regression