Cross-validation prior choice in Bayesian probit regression with many covariates

This paper examines prior choice in probit regression through a predictive cross-validation criterion. In particular, we focus on situations where the number of potential covariates is far larger than the number of observations, such as in gene expression data. Cross-validation avoids the tendency of such models to fit perfectly. We choose the scale parameter c in the standard variable selection prior as the minimizer of the log predictive score. Naive evaluation of the log predictive score requires substantial computational effort, and we investigate computationally cheaper methods using importance sampling.We find that K-fold importance densities perform best, in combination with either mixing over different values of c or with integrating over c through an auxiliary distribution.