Estimation and Conditional Inference in High-Dimensional Statistical Models

Abstract

In many areas of biology, recent advances in technology have facilitated the measurement of large numbers of features, while the number of observations in a data set may remain relatively modest. In this setting, lasso regression and related procedures have been extensively studied for prediction, while the problem of inference is relatively less studied. Most inference in high dimensions is based on simple marginal associations between variables. However, a richer characterization of the associations between variables can be obtained by examining conditional relationships, which account for the joint behavior of the variables. Inference on conditional relationships is more difficult, because it requires one to specify how features are related to one another, to estimate these relationships, and to characterize the uncertainty in the estimation procedure. In Chapters 2 and 3, we explore a few methods for testing hypotheses about conditional relationships in the high-dimensional setting. In Chapter 4, we note some strong distributional assumptions implicit in many treatments of high-dimensional graphical models, and propose a modification which treats this issue.