Selective Inference and Learning Mixed Graphical Models

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.

arXiv commentJason D. Lee PhD Dissertation

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• Pathwise coordinate optimization
• Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
• Valid post-selection inference
• High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
• Optimal Inference After Model Selection
• On Graphical Models via Univariate Exponential Family Distributions
• Exact Post-Selection Inference for Sequential Regression Procedures
• Tests in adaptive regression via the Kac-Rice formula
• High-dimensional Mixed Graphical Models
• Learning mixed graphical models from data with p larger than n