We introduce a natural conjugate prior for the transition matrix of a reversible Markov chain. This allows estimation and testing. The prior arises from random walk with reinforcement in the same way the Dirichlet prior arises from P\'{o}lya's urn. We give closed form normalizing constants, a simple method of simulation from the posterior and a characterization along the lines of W. E. Johnson's characterization of the Dirichlet prior.Comment: Published at http://dx.doi.org/10.1214/009053606000000290 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

We prove that the edge-reinforced random walk on the ladder ${\mathbb{Z}\times\{1,2\}}$ with initial weights $a>3/4$ is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edge-reinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece.Comment: Published at http://dx.doi.org/10.1214/009117905000000396 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from $\mathbb{Z}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that the linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point.Comment: Published in at http://dx.doi.org/10.1214/08-AOP446 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Let G be a finite tree. It is shown that edge-reinforced random walk on Z × G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced random walk on {0, 1, . . . , n} × G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance from the starting level. The estimates are uniform in n. They are used in the recurrence proof.

We give a characterization of a modi ed edge-reinforced random walk in terms of certain partially exchangeable sequences. In particular, we obtain a characterization of edge-reinforced random walk (introduced by Coppersmith and Diaconis) on a 2-edge-connected graph. Modifying the notion of partial exchangeability i n troduced by D i a c onis and Freedman in 2], we c haracterize unique mixtures of reversible Markov c hains under a recurrence assumption.