  We develop a forward-reverse EM (FREM) algorithm for estimating parameters
that determine the dynamics of a discrete time Markov chain evolving through a
certain measurable state space. As a key tool for the construction of the FREM
method we develop forward-reverse representations for Markov chains conditioned
on a certain terminal state. These representations may be considered as an
extension of the earlier work Bayer and Schoenmakers [2013] on conditional
diffusions. We proof almost sure convergence of our algorithm for a Markov
chain model with curved exponential family structure. On the numerical side we
give a complexity analysis of the forward-reverse algorithm by deriving its
expected cost. Two application examples are discuss to demonstrate the scope of
possible applications ranging from models based on continuous time processes to
discrete time Markov chain models.
