Selective Inference and Learning Mixed Graphical Models

TL;DR

This thesis advances selective inference methods for hypothesis testing after data-driven selection and introduces a new approach for learning the structure of mixed graphical models involving both continuous and discrete variables.

Contribution

It develops the Condition-on-Selection method for valid selective inference and proposes a novel symmetric group-lasso based approach for structure learning in mixed graphical models.

Findings

Condition-on-Selection method enables valid inference post-selection

New mixed graphical model with structure learning guarantees

Estimator is consistent in high-dimensional settings

Abstract

This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parametrization of the model. We provide conditions under which our estimator is model selection consistent in the high-dimensional regime.