Robust Multi-Hypothesis Testing with Moment Constrained Uncertainty Sets

TL;DR

This paper develops robust binary hypothesis tests that account for distributional uncertainty using moment constraints, providing optimal and tractable solutions for finite and infinite alphabets with demonstrated numerical performance.

Contribution

It introduces a novel robust testing framework based on moment-constrained uncertainty sets, including optimal, approximate, and generalized tests for various alphabet sizes.

Findings

Optimal test for finite alphabets derived.

Tractable approximation for infinite alphabets proposed.

Numerical results validate the robustness and effectiveness.

Abstract

The problem of robust binary hypothesis testing is studied. Under both hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through moments; in particular, the sets contain distributions whose moments are centered around the empirical moments obtained from training samples. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., minimize the worst-case error probability over the uncertainty sets. In the finite-alphabet case, the optimal test is obtained. In the infinite-alphabet case, a tractable approximation to the worst-case error is derived that converges to the optimal value using finite samples from the alphabet. A test is further constructed to generalize to the entire alphabet. An exponentially consistent test for testing batch samples is also proposed. Numerical results are provided to demonstrate the performance of the proposed robust tests.