  We present two alternative ways to apply PAC-Bayesian analysis to sequences
of dependent random variables. The first is based on a new lemma that enables
to bound expectations of convex functions of certain dependent random variables
by expectations of the same functions of independent Bernoulli random
variables. This lemma provides an alternative tool to Hoeffding-Azuma
inequality to bound concentration of martingale values. Our second approach is
based on integration of Hoeffding-Azuma inequality with PAC-Bayesian analysis.
We also introduce a way to apply PAC-Bayesian analysis in situation of limited
feedback. We combine the new tools to derive PAC-Bayesian generalization and
regret bounds for the multiarmed bandit problem. Although our regret bound is
not yet as tight as state-of-the-art regret bounds based on other
well-established techniques, our results significantly expand the range of
potential applications of PAC-Bayesian analysis and introduce a new analysis
tool to reinforcement learning and many other fields, where martingales and
limited feedback are encountered.
