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.

Label shift, a prevalent challenge in supervised learning, arises when the class prior distribution of test data differs from that of training data, leading to significant degradation in classifier performance. To accurately estimate the test priors and enhance classification accuracy, we propose a Bayesian framework for label shift estimation, termed Full Maximum A Posterior Label Shift (FMAPLS), along with its online version, online-FMAPLS. Leveraging batch and online Expectation-Maximization (EM) algorithms, these methods jointly and dynamically optimize Dirichlet hyperparameters $\boldsymbolα$ and class priors $\boldsymbolπ$, thereby overcoming the rigid constraints of the existing Maximum A Posterior Label Shift (MAPLS) approach. Moreover, we introduce a linear surrogate function (LSF) to replace gradient-based hyperparameter updates, yielding closed-form solutions that reduce computational complexity while retaining asymptotic equivalence. The online variant substitutes the batch E-step with a stochastic approximation, enabling real-time adaptation to streaming data. Furthermore, our theoretical analysis reveals a fundamental trade-off between online convergence rate and estimation accuracy. Extensive experiments on CIFAR100 and ImageNet datasets under shuffled long-tail and Dirichlet test priors demonstrate that FMAPLS and online-FMAPLS respectively achieve up to 40% and 12% lower KL divergence and substantial improvements in post-shift accuracy over state-of-the-art baselines, particularly under severe class imbalance and distributional uncertainty. These results confirm the robustness, scalability, and suitability of the proposed methods for large-scale and dynamic learning scenarios.

Summary and Contributions: This paper studies the label estimation problem and unifies a previously proposed perspective--maximum likelihood and calibration-- with the recent method using black-box predictors. The main takeaway is that these two perspectives can be regarded as the same framework and the calibration is a necessary step to achieve better performance. In the analysis, the weight estimation error is analyzed by decomposing it into an estimation error due to finite samples and an calibration error due to label shift. The empirical evaluation demonstrates that the combined method (MLLS with confusion matrix) outperforms using only black-box predictors. I keep my score after reading it.

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.