A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations π π i P i j = π π j P j i , where P ij is the Markov transition probability from state i to state j, i.e. P ij = P (X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively. When Pr(X t −1 = i) = π i for all i, this is equivalent to the joint probability matrix, Pr(X t −1 = i, X t = j) being symmetric in i and j ; or symmetric in t − 1 and t.