The problem of estimating a finite state Markov chain observed via a process on the same state space is discussed. Optimal solutions are given for both the "weak" and "strong" formulations of the problem. The "weak" formulation proceeds using a reference probability and a measure change for the Markov chain. The "strong" formulation considers an observation process related to perturbations of the counting processes associated with the Markov chain. In this case the "small noise" convergence is investigated.