  Rates of binomial processes are modeled using beta-binomial distributions
(for example, from Beta Regression). We treat the offline optimization scenario
and then the online one, where we optimize the exploration-exploitation
problem. The rates given by two processes are compared through their
distributions, but we would like to optimize the net payout (given a constant
value per successful event, unique for each of the processes). The result is an
analytically-closed, probabilistic, hypergeometric expression for comparing the
payout distributions of two processes. To conclude, we contrast this Bayesian
result with an alternative frequentist approach and find 4.5 orders of
magnitude improvement in performance, for a numerical accuracy level of 0.01%.
