  Atar, Chowdhary and Dupuis have recently exhibited a variational formula for
exponential integrals of bounded measurable functions in terms of R\'enyi
divergences. We develop a variational characterization of the R\'enyi
divergences between two probability distributions on a measurable sace in terms
of relative entropies. When combined with the elementary variational formula
for exponential integrals of bounded measurable functions in terms of relative
entropy, this yields the variational formula of Atar, Chowdhary and Dupuis as a
corollary. We also develop an analogous variational characterization of the
R\'enyi divergence rates between two stationary finite state Markov chains in
terms of relative entropy rates. When combined with Varadhan's variational
characterization of the spectral radius of square matrices with nonnegative
entries in terms of relative entropy, this yields an analog of the variational
formula of Atar, Chowdary and Dupuis in the framework of finite state Markov
chains.
