A Generalized Poor-Verdu Error Bound for Multihypothesis Testing and the Channel Reliability Function

TL;DR

This paper introduces a generalized lower bound on multihypothesis testing error probability, extending Poor and Verdu's earlier work, and applies it to derive tight bounds on the channel reliability function for certain channels.

Contribution

It generalizes the Poor-Verdu error bound using tilted posterior distributions and applies it to establish new bounds on the channel reliability function, including a multi-letter asymptotic expression.

Findings

The new bound generalizes previous bounds and can be exact in the limit of large tilting parameter.

One of the derived bounds is tight for certain channels, including the finite-input memoryless Gaussian channel.

Numerical examples demonstrate the bounds' effectiveness and limitations.

Abstract

A lower bound on the minimum error probability for multihypothesis testing is established. The bound, which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation with tilting parameter theta larger than or equal to 1, generalizes an earlier bound due the Poor and Verdu (1995). A sufficient condition is established under which the new bound (minus a multiplicative factor) provides the exact error probability in the limit of theta going to infinity. Examples illustrating the new bound are also provided. The application of this generalized Poor-Verdu bound to the channel reliability function is next carried out, resulting in two information-spectrum upper bounds. It is observed that, for a class of channels including the finite-input memoryless Gaussian channel, one of the bounds is tight and gives a multi-letter asymptotic expression for the reliability function, albeit its determination or calculation in single-letter form remains an open challenging problem. Numerical examples regarding the other bound are finally presented.