Factorizable joint shift revisited

TL;DR

This work addresses distribution shift under partial labels by extending factorable joint shift (FJS) to general label spaces, enabling both classification and regression under domain adaptation. It formalizes a density-based link between source and target distributions via $dQ/dP=f$, and shows that FJS can be written as $dQ/dP=\overline{h}(X)\overline{g}(Y)$, connecting covariate and label shifts in a unified framework. The authors derive a posterior-correction formula for $Q_{Y|X}$, discuss representations of $Q_X$ and $Q_Y$ through $h$ and $g$, and propose general strategies to recover $\overline{h}$, $\overline{g}$ (including an EM-like approach) from marginal target information, as well as an iterative method for GLS-based inference. They further show that GLS under a sufficient representation implies FJS, bridging representation learning with joint-shift analysis and highlighting practical pathways for domain adaptation with general label spaces.

Abstract

Factorizable joint shift (FJS) was proposed as a type of distribution shift (or dataset shift) that comprises both covariate and label shift. Recently, it has been observed that FJS actually arises from consecutive label and covariate (or vice versa) shifts. Research into FJS so far has been confined to the case of categorical label spaces. We propose a framework for analysing distribution shift in the case of general label spaces, thus covering both classification and regression models. Based on the framework, we generalise existing results on FJS to general label spaces and propose a related extension of the expectation maximisation (EM) algorithm for class prior probabilities. We also take a fresh look at generalized label shift (GLS) in the case of general label spaces.

Theorems & Definitions (27)

• Definition 2.1: Regular conditional distribution
• Proposition 2.3
• proof : Proof of Proposition \ref{['pr:genProperties']}
• Corollary 2.5
• Theorem 2.6
• proof : Proof of Theorem \ref{['th:condDens']}
• Corollary 2.7
• proof : Proof of Corollary \ref{['co:main']}.
• Proposition 2.10
• proof : Proof of Proposition \ref{['pr:condDist']}