Bounding Neyman-Pearson Region with $f$-Divergences

TL;DR

This paper introduces new bounds on the Neyman-Pearson region using $f$-divergences, improving existing inequalities and providing methods to construct distribution pairs that realize these bounds.

Contribution

It establishes the first bounds on the Neyman-Pearson boundary in terms of $f$-divergences, including a refined upper bound with the Chernoff coefficient, and offers methods to construct distribution pairs matching these bounds.

Findings

The bounds are tight and characterize the Neyman-Pearson boundary.

The KL divergence bound improves Pinsker's inequality.

A closed-form upper bound using the Chernoff coefficient is derived.

Abstract

The Neyman-Pearson region of a simple binary hypothesis testing is the set of points whose coordinates represent the false positive rate and false negative rate of some test. The lower boundary of this region is given by the Neyman-Pearson lemma, and is up to a coordinate change, equivalent to the optimal ROC curve. We establish a novel lower bound for the boundary in terms of any $f$-divergence. Since the bound generated by hockey-stick $f$-divergences characterizes the Neyman-Pearson boundary, this bound is best possible. In the case of KL divergence, this bound improves Pinsker's inequality. Furthermore, we obtain a closed-form refined upper bound for the Neyman-Pearson boundary in terms of the Chernoff $\alpha$-coefficient. Finally, we present methods for constructing pairs of distributions that can approximately or exactly realize any given Neyman-Pearson boundary.