A Unified View of Label Shift Estimation

Label shift describes the setting where although the label distribution might change between the source and target domains, the class-conditional probabilities (of data given a label) do not. There are two dominant approaches for estimating the label marginal. BBSE, a moment-matching approach based on confusion matrices, is provably consistent and provides interpretable error bounds. However, a maximum likelihood estimation approach, which we call MLLS, dominates empirically. In this paper, we present a unified view of the two methods and the first theoretical characterization of the likelihood-based estimator. Our contributions include (i) conditions for consistency of MLLS, which include calibration of the classifier and a confusion matrix invertibility condition that BBSE also requires; (ii) a unified view of the methods, casting the confusion matrix as roughly equivalent to MLLS for a particular choice of calibration method; and (iii) a decomposition of MLLS's finite-sample error into terms reflecting the impacts of miscalibration and estimation error. Our analysis attributes BBSE's statistical inefficiency to a loss of information due to coarse calibration. We support our findings with experiments on both synthetic data and the MNIST and CIFAR10 image recognition datasets.