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 $P(Y)\ne Q(Y)$. 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.