We define a conjugate prior for the reversible Markov chain of order r. The prior arises from a partially exchangeable reinforced random walk, in the same way that the Beta distribution arises from the exchangeable Polyá urn. An extension to variable-order Markov chains is also derived. We show the utility of this prior in testing the order and estimating the parameters of a reversible Markov model.
"Bayesian analysis of variable-order, reversible Markov chains." Ann. Statist. 39 (2) 838 - 864, April 2011. https://doi.org/10.1214/10-AOS857