  This note gives a short, self-contained, proof of a sharp connection between
Gittins indices and Bayesian upper confidence bound algorithms. I consider a
Gaussian multi-armed bandit problem with discount factor $\gamma$. The Gittins
index of an arm is shown to equal the $\gamma$-quantile of the posterior
distribution of the arm's mean plus an error term that vanishes as $\gamma\to
1$. In this sense, for sufficiently patient agents, a Gittins index measures
the highest plausible mean-reward of an arm in a manner equivalent to an upper
confidence bound.
