Under label shift, the label distribution p(y) might change but the
class-conditional distributions p(x|y) 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 MLLS. Our
contributions include (i) consistency conditions for MLLS, which include
calibration of the classifier and a confusion matrix invertibility condition
that BBSE also requires; (ii) a unified framework, casting BBSE 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
miscalibration and estimation error. Our analysis attributes BBSE's statistical
inefficiency to a loss of information due to coarse calibration. Experiments on
synthetic data, MNIST, and CIFAR10 support our findings.

研究标签移位问题下，使用 moment-matching 方法通过混淆矩阵来估算标签边际分布并提供可解释的误差界限，同时介绍了 MLLS 方法，并提出了一种统一方法来比较两种方法，并将 MLLS 的有限样本误差分解成反映误差估计和标签错误率校准不当的项，最后通过实验验证了其有效性。