Robust Label Shift Quantification

In this paper, we investigate the label shift quantification problem. We propose robust estimators of the label distribution which turn out to coincide with the Maximum Likelihood Estimator. We analyze the theoretical aspects and derive deviation bounds for the proposed method, providing optimal guarantees in the well-specified case, along with notable robustness properties against outliers and contamination. Our results provide theoretical validation for empirical observations on the robustness of Maximum Likelihood Label Shift.

💡 Research Summary

The paper addresses the problem of label‑shift quantification, i.e., estimating the target label distribution β* when the conditional distribution of covariates given the label remains unchanged between source and target domains. The authors propose robust estimators that coincide with the maximum‑likelihood estimator (MLE) but enjoy stronger theoretical guarantees under contamination and outlier scenarios.

The work is organized around two settings. In Setting A the conditional distributions Q₁,…,Qₖ are either known from the source data or are replaced by estimates Q₁,…,Qₖ. The goal is to recover the mixing weights β in the mixture model P* = ∑ₖ βᵢ Qᵢ. In Setting B the practitioner possesses a Bayes predictor f* trained on the source domain (or an estimate of it) and uses its probabilistic outputs to perform the same weight estimation. Both settings ultimately require estimating β∈Δₖ, the simplex of label probabilities.

The methodological core is the use of ρ‑estimators, a robust estimation framework introduced by Baraud et al. (2016, 2018). Instead of the usual log‑likelihood ratio, the authors employ a bounded transformation ψ(t) = (t − 1)/(t + 1) to construct a test statistic T(x,q,q′) = ∑ᵢ ψ(q′(Xᵢ)/q(Xᵢ)). For a candidate density q, the supremum of T over all alternatives q′ defines a criterion Υ(x,q). A ρ‑estimator is any density that nearly minimizes Υ. This construction yields estimators that are well‑defined even when the model densities are unbounded or when the true distribution is not absolutely continuous with respect to the model—a situation where classical MLE may fail.

A key theoretical contribution (Proposition 3.1) shows that whenever the MLE exists, it is also a ρ‑estimator for the mixture model Mₘᵢₓ(q₁,…,qₖ). Consequently, all the robustness results derived for ρ‑estimators automatically apply to the MLE, and standard algorithms such as EM can be used to compute the estimator in practice.

The authors derive non‑asymptotic deviation bounds measured in Hellinger distance. Under the identifiability condition that the component distributions Q₁,…,Qₖ are linearly independent, they prove that with probability at least 1 − δ, the Hellinger error satisfies
 h²(P*, ĤP) ≤ C