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

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) neq 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-B3SE), 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-B3SE through information geometry, showing that it generalizes existing shift estimators.