  In the stochastic multi-armed bandit problem, a randomized probability
matching policy called Thompson sampling (TS) has shown excellent performance
in various reward models. In addition to the empirical performance, TS has been
shown to achieve asymptotic problem-dependent lower bounds in several models.
However, its optimality has been mainly addressed under light-tailed or
one-parameter models that belong to exponential families. In this paper, we
consider the optimality of TS for the Pareto model that has a heavy tail and is
parameterized by two unknown parameters. Specifically, we discuss the
optimality of TS with probability matching priors that include the Jeffreys
prior and the reference priors. We first prove that TS with certain probability
matching priors can achieve the optimal regret bound. Then, we show the
suboptimality of TS with other priors, including the Jeffreys and the reference
priors. Nevertheless, we find that TS with the Jeffreys and reference priors
can achieve the asymptotic lower bound if one uses a truncation procedure.
These results suggest carefully choosing noninformative priors to avoid
suboptimality and show the effectiveness of truncation procedures in TS-based
policies.
