Bayesian M-ary Hypothesis Testing: The Meta-Converse and Verd\'u-Han Bounds are Tight

TL;DR

This paper derives two exact formulas for the minimum error probability in Bayesian M-ary hypothesis testing, demonstrating the tightness of key converse bounds and enhancing understanding of their limitations.

Contribution

It introduces two novel exact characterizations of the minimum error probability, establishing the tightness of the meta-converse and Verdú-Han bounds in information theory.

Findings

Proves the tightness of the meta-converse bound.

Shows the tightness of the Verdú-Han lower bound.

Provides formulas that identify where existing bounds are loose.

Abstract

Two alternative exact characterizations of the minimum error probability of Bayesian M-ary hypothesis testing are derived. The first expression corresponds to the error probability of an induced binary hypothesis test and implies the tightness of the meta-converse bound by Polyanskiy, Poor and Verd\'u; the second expression is function of an information-spectrum measure and implies the tightness of a generalized Verd\'u-Han lower bound. The formulas characterize the minimum error probability of several problems in information theory and help to identify the steps where existing converse bounds are loose.