Bayesian analysis of variable-order, reversible Markov chains

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 Poly'{a} 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.