Risk awareness is an important feature to formulate a variety of real world problems. In this paper we study a multi-arm bandit problem in which the quality of each arm is measured by the Conditional Value at Risk (CVaR) at some level {\alpha} of the reward distribution. While existing works in this setting mainly focus on Upper Confidence Bound algorithms, we introduce the first Thompson Sampling approaches for CVaR bandits. Building on a recent work by Riou and Honda (2020), we propose {\alpha}-NPTS for bounded rewards and {\alpha}-Multinomial-TS for multinomial distributions. We provide a novel lower bound on the CVaR regret which extends the concept of asymptotic optimality to CVaR bandits and prove that {\alpha}-Multinomial-TS is the first algorithm to achieve this lower bound. Finally, we demonstrate empirically the benefit of Thompson Sampling approaches over their UCB counterparts.
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