  The empirically successful Thompson Sampling algorithm for stochastic bandits
has drawn much interest in understanding its theoretical properties. One
important benefit of the algorithm is that it allows domain knowledge to be
conveniently encoded as a prior distribution to balance exploration and
exploitation more effectively. While it is generally believed that the
algorithm's regret is low (high) when the prior is good (bad), little is known
about the exact dependence. In this paper, we fully characterize the
algorithm's worst-case dependence of regret on the choice of prior, focusing on
a special yet representative case. These results also provide insights into the
general sensitivity of the algorithm to the choice of priors. In particular,
with $p$ being the prior probability mass of the true reward-generating model,
we prove $O(\sqrt{T/p})$ and $O(\sqrt{(1-p)T})$ regret upper bounds for the
bad- and good-prior cases, respectively, as well as \emph{matching} lower
bounds. Our proofs rely on the discovery of a fundamental property of Thompson
Sampling and make heavy use of martingale theory, both of which appear novel in
the literature, to the best of our knowledge.
