Selective Inference and Learning Mixed Graphical Models
It provides methods for valid inference after data selection and extends graphical model learning to mixed data types, though it is incremental as it generalizes existing approaches.
The thesis addresses two statistical problems: selective inference for data-driven hypotheses, with a focus on lasso-selected regression coefficients, and structure learning of mixed graphical models for continuous and discrete variables, achieving model selection consistency under high-dimensional conditions.
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.