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This paper gives upper and lower bounds on the minimum error probability ofBayesian $M$-ary hypothesis testing in terms of the Arimoto-R\'enyi conditionalentropy of an arbitrary order $\alpha$. The improved tightness of these boundsover their specialized versions with the Shannon conditional entropy($\alpha=1$) is demonstrated. In particular, in the case where $M$ is finite,we show how to generalize Fano's inequality under both the conventional andlist-decision settings. As a counterpart to the generalized Fano's inequality,allowing $M$ to be infinite, a lower bound on the Arimoto-R\'enyi conditionalentropy is derived as a function of the minimum error probability. Explicitupper and lower bounds on the minimum error probability are obtained as afunction of the Arimoto-R\'enyi conditional entropy for both positive andnegative $\alpha$. Furthermore, we give upper bounds on the minimum errorprobability as functions of the R\'enyi divergence. In the setup of discretememoryless channels, we analyze the exponentially vanishing decay of theArimoto-R\'enyi conditional entropy of the transmitted codeword given thechannel output when averaged over a random coding ensemble.

Arimoto-R\'enyi Conditional Entropy and Bayesian $M$-ary Hypothesis Testing