Title: Several Studies in Label Shift

Abstract: We provide an introduction to label shift problems. In the context of discrete response, we study the importance weights confidence set problem by a paradigm shift from traditional inversion-based inference to a direct matrix constraint framework. We use this framework to characterize a joint confidence region and extract marginal intervals via linear programming, deriving provably tighter bounds for importance weights while maintaining exact finite-sample validity. In the context of continuous response, we study the estimation and inference of a general target population characteristric by developing doubly and singly robust estimators as well as the efficient estimator. Many ongoing and future developments will be discussed too.