The Bayesian bootstrap for Markov chains is the Bayesian analogue of the bootstrap method for Markov chains. We construct a random-weighted empirical distribution, based on i.i.d. exponential random variables, to simulate the posterior distribution of the transition probability, the stationary probability, as well as the first hitting time up to a specific state, of a finite state ergodic Markov chain. The large sample theory is developed which shows that with a matrix beta prior on the transition probability, the Bayesian bootstrap procedure is second-order consistent for approximating the pivot of the posterior distributions of the transition probability. The small sample properties of the Bayesian bootstrap are also discussed by a simulation study.
