  Thompson Sampling provides an efficient technique to introduce prior
knowledge in the multi-armed bandit problem, along with providing remarkable
empirical performance. In this paper, we revisit the Thompson Sampling
algorithm under rewards drawn from symmetric $\alpha$-stable distributions,
which are a class of heavy-tailed probability distributions utilized in finance
and economics, in problems such as modeling stock prices and human behavior. We
present an efficient framework for posterior inference, which leads to two
algorithms for Thompson Sampling in this setting. We prove finite-time regret
bounds for both algorithms, and demonstrate through a series of experiments the
stronger performance of Thompson Sampling in this setting. With our results, we
provide an exposition of symmetric $\alpha$-stable distributions in sequential
decision-making, and enable sequential Bayesian inference in applications from
diverse fields in finance and complex systems that operate on heavy-tailed
features.
