  This paper gives upper and lower bounds on the minimum error probability of
Bayesian $M$-ary hypothesis testing in terms of the Arimoto-R\'enyi conditional
entropy of an arbitrary order $\alpha$. The improved tightness of these bounds
over 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 and
list-decision settings. As a counterpart to the generalized Fano's inequality,
allowing $M$ to be infinite, a lower bound on the Arimoto-R\'enyi 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-R\'enyi conditional entropy for both positive and
negative $\alpha$. Furthermore, we give upper bounds on the minimum error
probability as functions of the R\'enyi divergence. In the setup of discrete
memoryless channels, we analyze the exponentially vanishing decay of the
Arimoto-R\'enyi conditional entropy of the transmitted codeword given the
channel output when averaged over a random coding ensemble.
