Relations among the Minimum Error Probability, Guessing Moments, and Arimoto-Rényi Conditional Entropy

Igal Sason, Sergio Verdú

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

This talk presents upper and lower bounds on the minimum error probability of
Bayesian M-ary hypothesis testing in terms of the Arimoto-Rényi conditional entropy
of an arbitrary order α. The improved tightness of these bounds over their specialized
versions with the Shannon conditional entropy (α = 1) is explained. In particular, in the
case where M is finite, we generalize Fano’s inequality under both the conventional and
list-decision settings. As a counterpart to the generalized Fano’s inequality, allowing M
to be infinite, a lower bound on the Arimoto-Rényi conditional entropy is derived as a
function of the minimum error probability. We further provide upper and lower bounds on
the optimal guessing moments of a random variable taking values on a finite set when side
information may be available. These moments quantify the number of guesses required
for correctly identifying the unknown object and, similarly to Arıkan’s bounds, they are
expressed in terms of the Arimoto-Rényi conditional entropy. Although Arıkan’s bounds
are asymptotically tight, the improvement of the bounds in this paper is significant in the
non-asymptotic regime. Relationships between moments of the optimal guessing function
and the MAP error probability are also presented, characterizing the exact locus of their
attainable values.
Original languageEnglish
Title of host publicationInternational Zurich Seminar on Information and Communication
Subtitle of host publicationIZS 2018
DOIs
StatePublished - 2018
EventInternational Zurich Seminar on Information and Communication - Zurich
Duration: 21 Feb 201823 Feb 2018
https://www.research-collection.ethz.ch/handle/20.500.11850/242151

Conference

ConferenceInternational Zurich Seminar on Information and Communication
Abbreviated titleIZS
CityZurich
Period21/02/1823/02/18
Internet address

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