Graph-Smoothed Bayesian Black-Box Shift Estimator and Its Information Geometry

2025-05-22

AI Summary

Label shift adaptation aims to estimate the target-domain class prior $Q(Y)$ under the assumption that $P(X|Y) = Q(X|Y)$ but $P(Y) eq Q(Y)$. Conventional black-box estimators ignore sampling noise and inter-class similarity, yielding only fragile point estimates. To address this, we propose a graph-smoothed Bayesian framework: for the first time, we jointly impose Laplacian–Gaussian priors on the target log-prior and confusion matrix columns, leveraging a label similarity graph. We theoretically establish its $N^{-1/2}$ estimation consistency, algebraic connectivity–driven variance reduction, and robustness to Laplacian prior misspecification. Estimation is performed via Bayesian inference, graph Laplacian regularization, and Hamiltonian Monte Carlo—yielding low-variance, robust posterior estimates. Our framework unifies several existing methods under a single probabilistic interpretation. Empirically, it achieves significant improvements in both accuracy and stability on real-world label-shifted benchmarks.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

Abstract

Label shift adaptation aims to recover target class priors when the labelled source distribution $P$ and the unlabelled target distribution $Q$ share $P(X mid Y) = Q(X mid Y)$ but $P(Y) eq Q(Y)$. Classical black-box shift estimators invert an empirical confusion matrix of a frozen classifier, producing a brittle point estimate that ignores sampling noise and similarity among classes. We present Graph-Smoothed Bayesian BBSE (GS-B$^3$SE), a fully probabilistic alternative that places Laplacian-Gaussian priors on both target log-priors and confusion-matrix columns, tying them together on a label-similarity graph. The resulting posterior is tractable with HMC or a fast block Newton-CG scheme. We prove identifiability, $N^{-1/2}$ contraction, variance bounds that shrink with the graph's algebraic connectivity, and robustness to Laplacian misspecification. We also reinterpret GS-B$^3$SE through information geometry, showing that it generalizes existing shift estimators.

Problem

• Estimates target class priors under label shift
• Improves robustness over classical black-box shift estimators
• Uses graph-based priors for probabilistic confusion matrix modeling

Innovation

• Graph-smoothed Bayesian estimator for label shift
• Laplacian-Gaussian priors on target log-priors
• Tractable posterior with HMC or Newton-CG