Mixture Proportion Estimation and Weakly-supervised Kernel Test for Conditional Independence

TL;DR

This paper introduces new assumptions based on conditional independence for mixture proportion estimation, enabling identifiability without irreducibility, and develops kernel tests for validation in weakly supervised learning.

Contribution

It proposes novel CI-based assumptions for identifiability in MPE and develops associated estimators and tests, advancing weakly supervised learning methods.

Findings

Estimators outperform existing methods in accuracy.

Kernel tests effectively control error rates.

Assumptions enable identifiability without irreducibility.

Abstract

Mixture proportion estimation (MPE) aims to estimate class priors from unlabeled data. This task is a critical component in weakly supervised learning, such as PU learning, learning with label noise, and domain adaptation. Existing MPE methods rely on the \textit{irreducibility} assumption or its variant for identifiability. In this paper, we propose novel assumptions based on conditional independence (CI) given the class label, which ensure identifiability even when irreducibility does not hold. We develop method of moments estimators under these assumptions and analyze their asymptotic properties. Furthermore, we present weakly-supervised kernel tests to validate the CI assumptions, which are of independent interest in applications such as causal discovery and fairness evaluation. Empirically, we demonstrate the improved performance of our estimators compared with existing methods and that our tests successfully control both type I and type II errors.\label{key}