Hidden Markov models with state-dependent mixtures: minimal representation, model testing and applications to clustering

Let (St)t be a k-state Markov chain with ergodic transition probability matrix (t.p.m.) Γ = (γi,j)i,j=1,...,k having the stationary distribution π = (π1, . . . , πk). In the following we always assume πj > 0 for j = 1, . . . , k. For a (disjoint) partition G = {G1, . . . , Gr} of the state space (into non-empty sets), we let G(j) be the function which maps a state j ∈ {1, . . . , k} onto its group, i.e. for j ∈ Gl we have G(j) = Gl. If P ( St−1 = i, St ∈ G(j) ) > 0) we have the general formula