Maxentropy completion and properties of some partially defined Stationary Markov chains

We consider a stationary Markovian evolution with values on a disjointly partitioned set space I E. The evolution is visible (in the sense of knowing the transition probabilities) on the states in I but not for the states in E. One only knows some partial information on the transition probabilities on E, the input and output transition probabilities and some constraints of the transition probabilities on E. Under some conditions we supply the transition probabilities on E that satisfies the maximum entropy principle.