Renyi's entropy and the probability of error

Abstract

The basic properties of Renyi's entropy are reviewed, and its concavity properties are characterized. New bounds (referred to as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{\alpha}</tex> bounds) on the probability of error are derived from Renyi's entropy and are compared with known bounds. It is proved that for the two-class case, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{2}</tex> bound is sharper than many of the previously known bounds. The difference between the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{2}</tex> bound and the real value of the probability of error is at most 0.09.