  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.
