  Bayesian variable selection is a powerful tool for data analysis, as it
offers a principled method for variable selection that accounts for prior
information and uncertainty. However, wider adoption of Bayesian variable
selection has been hampered by computational challenges, especially in
difficult regimes with a large number of covariates P or non-conjugate
likelihoods. To scale to the large P regime we introduce an efficient MCMC
scheme whose cost per iteration is sublinear in P. In addition we show how this
scheme can be extended to generalized linear models for count data, which are
prevalent in biology, ecology, economics, and beyond. In particular we design
efficient algorithms for variable selection in binomial and negative binomial
regression, which includes logistic regression as a special case. In
experiments we demonstrate the effectiveness of our methods, including on
cancer and maize genomic data.
