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.
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 language | English |
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Title of host publication | International Zurich Seminar on Information and Communication |
Subtitle of host publication | IZS 2018 |
DOIs | |
State | Published - 2018 |
Event | International Zurich Seminar on Information and Communication - Zurich Duration: 21 Feb 2018 → 23 Feb 2018 https://www.research-collection.ethz.ch/handle/20.500.11850/242151 |
Conference
Conference | International Zurich Seminar on Information and Communication |
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Abbreviated title | IZS |
City | Zurich |
Period | 21/02/18 → 23/02/18 |
Internet address |