Approximate selective inference via maximum likelihood

This article proposes post-selective inference for Gaussian models via approximate maximum likelihood. Our proposal serves two key goals: (i) efficient utility of hold-out information from selection by exploiting randomization; (ii) computational ease by bypassing expensive MCMC samplers from intractable conditional distributions. At the core of our method is the solution to a convex optimization problem that assumes a separable form across multiple learning queries during selection. Our proposal allows us to tackle efficient and tractable inference in many practical scenarios where more than one query informs inference. We illustrate the potential of our approximate method in comparisons with existing strategies across wide ranging signal-to-noise regimes and on gene expression data from TCGA (The Cancer Genome Atlas).