Hypothesis Tests for Distributional Group Symmetry with Applications to Particle Physics

Abstract

Symmetry plays a central role in the sciences, machine learning, and statistics. When data are known to obey a symmetry, various methods that exploit symmetry have been developed. However, statistical tests for the presence of group invariance focus on a handful of specialized situations, and tests for equivariance are largely non-existent. This work formulates non-parametric hypothesis tests, based on a single independent and identically distributed sample, for distributional symmetry under a specified group. We provide a general formulation of tests for symmetry within two broad settings. Generalizing existing theory for group-based randomization tests, the first setting tests for the invariance of a marginal or joint distribution under the action of a compact group. The second setting tests for the invariance or equivariance of a conditional distribution under the action of a locally compact group. We show that the test for conditional symmetry can be formulated as a test for conditional independence. We implement our tests using kernel methods and apply them to testing for symmetry in problems from high-energy particle physics.