General Classes of Lower Bounds on the Probability of Error in Multiple Hypothesis Testing

TL;DR

This paper introduces two new classes of lower bounds on the probability of error in multiple hypothesis testing, derived using Holder's inequalities, which generalize existing bounds and closely approximate the optimal error probability.

Contribution

The paper proposes novel lower bounds based on Holder's inequalities, extending and generalizing existing bounds, and demonstrating their asymptotic tightness and computational advantages.

Findings

Bounds closely match the MAP error probability asymptotically.

New bounds outperform some existing bounds in tightness.

Bounds are computationally efficient for typical detection problems.

Abstract

In this paper, two new classes of lower bounds on the probability of error for $m$-ary hypothesis testing are proposed. Computation of the minimum probability of error which is attained by the maximum a-posteriori probability (MAP) criterion is usually not tractable. The new classes are derived using Holder's inequality and reverse Holder's inequality. The bounds in these classes provide good prediction of the minimum probability of error in multiple hypothesis testing. The new classes generalize and extend existing bounds and their relation to some existing upper bounds is presented. It is shown that the tightest bounds in these classes asymptotically coincide with the optimum probability of error provided by the MAP criterion for binary or multiple hypothesis testing problem. These bounds are compared with other existing lower bounds in several typical detection and classification problems in terms of tightness and computational complexity.