  We consider MCMC methods for learning equivalence classes of sparse Gaussian
DAG models when $p = e^{o(n)}$. The main contribution of this work is a rapid
mixing result for a random walk Metropolis-Hastings algorithm, which we prove
using a canonical path method. It reveals that the complexity of Bayesian
learning of sparse equivalence classes grows only polynomially in $n$ and $p$,
under some common high-dimensional assumptions. Further, a series of
high-dimensional consistency results is obtained by the path method, including
the strong selection consistency of an empirical Bayes model for structure
learning and the consistency of a greedy local search on the restricted search
space. Rapid mixing and slow mixing results for other structure-learning MCMC
methods are also derived. Our path method and mixing time results yield crucial
insights into the computational aspects of high-dimensional structure learning,
which may be used to develop more efficient MCMC algorithms.
