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We present here two algorithms based on Bayesian modelling of the MAB, that are proved to be optimal in term of frequentist regret. The first, Bayes-UCB uses a Bayesian Upper Confidence Bound based on quantiles of the posterior distribution as an index. The second is Thompson Sampling, a randomized Bayesian algorithm dating back to 1933 whose first finite-time regret analysis was proposed at COLT (2012) and for which we now have an optimal finite-time analysis.

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