An approximation-based framework for post-selective inference

This thesis discusses an approximation-based framework for post selective inference. Such approximations make it possible to bypass the intractability of randomized likelihoods in frequentist inference and posteriors formed by appending truncated likelihoods with priors in a Bayesian post-selective framework. The computational bottleneck in computing the conditional likelihood of data post randomized selection strategies is that after conditioning out nuisance parameters, such likelihoods no longer reduce to univariate truncated Gaussian laws as in Lee et al. (2016). In fact, the conditional likelihood with a randomized response is mostly not available in closed form expressions, thereby, demanding tools to make inference based on such selection-modified laws tractable. Intractability of the conditional likelihood, again poses major computational hurdles while providing inference based on a Bayesian model after mining the data where exploration may allow the analyst to discover parameterizations of interest and elicit plausible models on the joint space of data and parameters. Adopting Yekutieli (2012) ideas, where a Bayesian model post selection consists of a prior and a truncated likelihood, the resulting posterior distribution is affected by the very fact that selection was applied unlike in the setup usually considered when performing Bayesian variable selection. At the core of the methodology introduced in this thesis is a convex approximation to the truncated likelihood, which facilitates sampling from an approximate adjusted posterior distribution to provide Bayesian inference post selection and allows frequentist inference by a grid approximation to the conditional law of the target statistics, after eliminating nuisance parameters. Prior works in selective inference focus mainly on hypothesis testing, and capitalize on reductions achieved by conditioning out nuisance parameters. However, the techniques developed in that venue are generally less appropriate for addressing other questions, like point estimation. On the other hand, relying on an approximation to the full truncated likelihood, the tools we develop allow for more versatility including the computation of the maximum likelihood estimator (MLE). The approximation to the intractable normalizer in the conditional likelihood leads to a convex approximation to the truncated likelihood, which makes it computationally easy to numerically optimize and to analyze. The guarantee associated with the proposed approximation to the normalizer is that it captures the large deviations rate of decay of the exact selection probability. In fact, replacing the genuine truncated likelihood by its approximation, we can approximate the maximum-likelihood estimate (MLE) by solving a convex optimization problem. In fact, the MLE is globally consistent after selection, that is, it converges to the target population parameter in probability when conditioned on selection. The work in this thesis develops methodologies based on these approximation tool boxes and applies them to various real data settings to explore their use in different applications.

Keywords: Statistics.
Advisor: Taylor Jonathan E. 
Sabatti Chiara 
Tibshirani Robert