Simulation-based Regularized Logistic Regression

Abstract

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 $p\gg n$ 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.