  The aim of this work is to provide bounds connecting two probability measures
of the same event using R\'enyi $\alpha$-Divergences and Sibson's
$\alpha$-Mutual Information, a generalization of respectively the
Kullback-Leibler Divergence and Shannon's Mutual Information. A particular case
of interest can be found when the two probability measures considered are a
joint distribution and the corresponding product of marginals (representing the
statistically independent scenario). In this case, a bound using Sibson's
$\alpha-$Mutual Information is retrieved, extending a result involving Maximal
Leakage to general alphabets. These results have broad applications, from
bounding the generalization error of learning algorithms to the more general
framework of adaptive data analysis, provided that the divergences and/or
information measures used are amenable to such an analysis ({\it i.e.,} are
robust to post-processing and compose adaptively). The generalization error
bounds are derived with respect to high-probability events but a corresponding
bound on expected generalization error is also retrieved.
