Recursive Smoothers for Hidden Discrete-time Markov Chains

Hidden Markov chains have been the subject of extensive studies, see the books [1, 2] and the references therein. Of particular interest are the discret-time, finite-state hidden Markov models. In this paper, using the same techniques as in [3], we propose results that improve the finite-dimensional smoothers of functionals of a partially observed discrete-time Markov chain. The model itself extends models discussed in [2]. The proposed formulae for updating these quantities are recursive. Therefore, recalculation of all backward estimates is not required in the implementation of the EM algorithm. This paper is organized as follows. In Section 2, we introduce the model. In Section 3, a new probability measure under which all processes are independent is defined and a recursive filter for the state is derived. The main results of this paper are in Section 4 where recursive smoothers are derived.