  Several strategies have been developed recently to ensure valid inference
after model selection; some of these are easy to compute, while others fare
better in terms of inferential power. In this paper, we consider a selective
inference framework for Gaussian data. We propose a new method for inference
through approximate maximum likelihood estimation. Our goal is to: (i) achieve
better inferential power with the aid of randomization, (ii) bypass expensive
MCMC sampling from exact conditional distributions that are hard to evaluate in
closed forms. We construct approximate inference, e.g., p-values, confidence
intervals etc., by solving a fairly simple, convex optimization problem. We
illustrate the potential of our method across wide-ranging values of
signal-to-noise ratio in simulations. On a cancer gene expression data set we
find that our method improves upon the inferential power of some commonly used
strategies for selective inference.
