5d69dc892ba6e79fda0c6a1e286f24c5-Supplemental.pdf

Consider any predictor cM( |i) (as a function of the sample pathX) for theith row ofM, i = 1,2,3. In Section 6.2.2, we make the steps in(29) precise and bound the Bayes risk from below by an appropriate mutual information. In Section 6.2.3, we choose a prior distribution on the transition probabilities and prove a lower bound on the resulting mutual information, thereby completing the proof ofTheorem 1,with the added bonus that the construction isrestricted toirreducible and reversiblechains. Let (X1,...,Xn) be the trajectory of a stationary Markov chain with transition matrixM. We first relate the Bayes estimator ofM and T (given the X and Y chain respectively).