We cast model-free reinforcement learning as the problem of maximizing the likelihood of a probabilistic mixture model via sampling, addressing both the innite and nite horizon cases. We describe a Stochastic Approximation EM algorithm for likelihood maximization that, in the tabular case, is equivalent to a non-bootstrapping optimistic policy iteration algorithm like Sarsa(1) that can be applied both in MDPs and POMDPs. On the theoretical side, by relating the proposed stochastic EM algorithm to the family of optimistic policy iteration algorithms, we provide new tools that permit the design and analysis of algorithms in that family. On the practical side, preliminary experiments on a POMDP problem demonstrated encouraging results.