Minimum Probability of Error of List M-ary Hypothesis Testing

TL;DR

This paper analyzes the minimum probability of error in list M-ary hypothesis testing, providing exact expressions that relate to binary hypothesis testing and likelihood ratios, advancing understanding of list-based decision strategies.

Contribution

It introduces two exact formulas for the minimum error probability in list hypothesis testing, connecting it to binary tests and likelihood ratio tail probabilities.

Findings

Derived an expression as the error probability of a non-Bayesian binary test.

Connected the minimum error to the tail probability of likelihood ratios.

Provided a theoretical framework for list hypothesis testing error analysis.

Abstract

We study a variation of Bayesian M-ary hypothesis testing in which the test outputs a list of L candidates out of the M possible upon processing the observation. We study the minimum error probability of list hypothesis testing, where an error is defined as the event where the true hypothesis is not in the list output by the test. We derive two exact expressions of the minimum probability or error. The first is expressed as the error probability of a certain non-Bayesian binary hypothesis test, and is reminiscent of the meta-converse bound. The second, is expressed as the tail probability of the likelihood ratio between the two distributions involved in the aforementioned non-Bayesian binary hypothesis test.