  We consider reinforcement learning in parameterized Markov Decision Processes
(MDPs), where the parameterization may induce correlation across transition
probabilities or rewards. Consequently, observing a particular state transition
might yield useful information about other, unobserved, parts of the MDP. We
present a version of Thompson sampling for parameterized reinforcement learning
problems, and derive a frequentist regret bound for priors over general
parameter spaces. The result shows that the number of instants where suboptimal
actions are chosen scales logarithmically with time, with high probability. It
holds for prior distributions that put significant probability near the true
model, without any additional, specific closed-form structure such as conjugate
or product-form priors. The constant factor in the logarithmic scaling encodes
the information complexity of learning the MDP in terms of the Kullback-Leibler
geometry of the parameter space.
