A discrete time, finite state Hidden Markov Model is considered. This consists of a Markov chain and a finite state observation process. A recent paper considered the case when the noises in the chain and the observation process are independent; this paper discusses the more complicated dependent case. Recursive conditional normalized and unnormalized, (Zakai), estimates are obtained for: the state of the chain, the number of jumps of the chain from one state to another, the occupation time of the chain in any state, the number of transitions of the observation process and the number of joint transitions of the chain and observation process. These results allow reestimation of the parameters of the model. In particular, they provide a test for whether the noise terms in the chain and observation are independent