Arimoto-Rényi conditional entropy and Bayesian hypothesis testing

This paper gives 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 demonstrated. In particular, in the case where M is finite, we show how to 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. Explicit upper and lower bounds on the minimum error probability are obtained as a function of the Arimoto-Rényi conditional entropy.