  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 or non-conjugate
likelihoods. Generalized linear models for count data, which are prevalent in
biology, ecology, economics, and beyond, represent an important special case.
Here we introduce an efficient MCMC scheme for variable selection in binomial
and negative binomial regression that exploits Tempered Gibbs Sampling (Zanella
and Roberts, 2019) and that includes logistic regression as a special case. In
experiments we demonstrate the effectiveness of our approach, including on
cancer data with seventeen thousand covariates.
