  Thompson Sampling has been demonstrated in many complex bandit models,
however the theoretical guarantees available for the parametric multi-armed
bandit are still limited to the Bernoulli case. Here we extend them by proving
asymptotic optimality of the algorithm using the Jeffreys prior for
1-dimensional exponential family bandits. Our proof builds on previous work,
but also makes extensive use of closed forms for Kullback-Leibler divergence
and Fisher information (and thus Jeffreys prior) available in an exponential
family. This allow us to give a finite time exponential concentration
inequality for posterior distributions on exponential families that may be of
interest in its own right. Moreover our analysis covers some distributions for
which no optimistic algorithm has yet been proposed, including heavy-tailed
exponential families.
