Randomization Tests for Conditional Group Symmetry

TL;DR

This paper introduces nonparametric randomization tests for conditional symmetry in distributions, providing a general framework with finite-sample error control and practical kernel-based implementations, validated through synthetic and physics data.

Contribution

It develops the first nonparametric tests for conditional symmetry, with a comprehensive framework and kernel methods for practical implementation and theoretical guarantees.

Findings

Tests control Type I error in finite samples

Kernel methods provide power bounds

Empirical validation on synthetic and physics data

Abstract

Symmetry plays a central role in the sciences, machine learning, and statistics. While statistical tests for the presence of distributional invariance with respect to groups have a long history, tests for conditional symmetry in the form of equivariance or conditional invariance are absent from the literature. This work initiates the study of nonparametric randomization tests for symmetry (invariance or equivariance) of a conditional distribution under the action of a specified locally compact group. We develop a general framework for randomization tests with finite-sample Type I error control and, using kernel methods, implement tests with finite-sample power lower bounds. We also describe and implement approximate versions of the tests, which are asymptotically consistent. We study their properties empirically using synthetic examples and applications to testing for symmetry in two problems from high-energy particle physics.