  In this paper, we develop a simulation-based framework for regularized
logistic regression, exploiting two novel results for scale mixtures of
normals. By carefully choosing a hierarchical model for the likelihood by one
type of mixture, and implementing regularization with another, we obtain new
MCMC schemes with varying efficiency depending on the data type (binary v.
binomial, say) and the desired estimator (maximum likelihood, maximum a
posteriori, posterior mean). Advantages of our omnibus approach include
flexibility, computational efficiency, applicability in p >> n settings,
uncertainty estimates, variable selection, and assessing the optimal degree of
regularization. We compare our methodology to modern alternatives on both
synthetic and real data. An R package called reglogit is available on CRAN.
