Global Testing Against Sparse Alternatives under Ising Models

TL;DR

This paper investigates how dependence affects the detection of sparse signals in binary data modeled by Ising models, revealing phase transition effects and proposing an optimal testing procedure.

Contribution

It characterizes the impact of dependence on signal detectability and introduces a broadly applicable, asymptotically minimax optimal testing method.

Findings

Dependence can enhance detection at critical points.

A new testing procedure is asymptotically minimax optimal.

Phase transition influences signal detectability.

Abstract

In this paper, we study the effect of dependence on detecting sparse signals. In particular, we focus on global testing against sparse alternatives for the means of binary outcomes following an Ising model, and establish how the interplay between the strength and sparsity of a signal determines its detectability under various notions of dependence. The profound impact of dependence is best illustrated under the Curie-Weiss model where we observe the effect of a "thermodynamic" phase transition. In particular, the critical state exhibits a subtle "blessing of dependence" phenomenon in that one can detect much weaker signals at criticality than otherwise. Furthermore, we develop a testing procedure that is broadly applicable to account for dependence and show that it is asymptotically minimax optimal under fairly general regularity conditions.