  We consider the stochastic multi-armed bandit problem with a prior
distribution on the reward distributions. We are interested in studying
prior-free and prior-dependent regret bounds, very much in the same spirit as
the usual distribution-free and distribution-dependent bounds for the
non-Bayesian stochastic bandit. Building on the techniques of Audibert and
Bubeck [2009] and Russo and Roy [2013] we first show that Thompson Sampling
attains an optimal prior-free bound in the sense that for any prior
distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This
result is unimprovable in the sense that there exists a prior distribution such
that any algorithm has a Bayesian regret bounded from below by $\frac{1}{20}
\sqrt{n K}$. We also study the case of priors for the setting of Bubeck et al.
[2013] (where the optimal mean is known as well as a lower bound on the
smallest gap) and we show that in this case the regret of Thompson Sampling is
in fact uniformly bounded over time, thus showing that Thompson Sampling can
greatly take advantage of the nice properties of these priors.
