Bayesian inference for high‐dimensional linear regression under mnet priors

For regression problems that involve many potential predictors, the Bayesian variable selection (BVS) method is a powerful tool. This method associates each model with its posterior probability and achieves excellent prediction performance through Bayesian model averaging. The main challenges of using such models include specifying a suitable prior and computing posterior quantities for inference. We contribute to the literature of BVS modelling in the following aspects. We first propose a new family of priors, called the mnet prior, which is indexed by a few hyperparameters that allow great flexibility in the prior density. The hyperparameters can also be treated as random, so that their values need not be tuned manually, but will instead adapt to the data. Simulation studies are used to demonstrate good prediction and variable selection performances of these models. Secondly, the analytical expression of the posterior distribution is unavailable for the BVS model under the mnet prior in general, as is the case for most BVS models. We develop an adaptive Markov chain Monte Carlo algorithm that facilitates the computation in high‐dimensional regression problems. We finally showcase various ways to do inference with BVS models, highlighting a new way to visualize the importance of each predictor along with estimation of the coefficients and their uncertainties. These are demonstrated through the analysis of a breast cancer gene expression dataset. The Canadian Journal of Statistics 44: 180–197; 2016 © 2016 Statistical Society of Canada