Maxentropy Completion and Properties of Some Partially Defined Stationary Markov Chains

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