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Structure learning via MCMC sampling is known to be very challenging becauseof the enormous search space and the existence of Markov equivalent DAGs.Theoretical results on the mixing behavior are lacking. In this work, we provethe rapid mixing of a random walk Metropolis-Hastings algorithm, which revealsthat the complexity of Bayesian learning of sparse equivalence classes growsonly polynomially in $n$ and $p$, under some high-dimensional assumptions. Aseries of high-dimensional consistency results is obtained, including thestrong selection consistency of an empirical Bayes model for structurelearning. Our proof is based on two new results. First, we derive a generalmixing time bound on finite state spaces, which can be applied to local MCMCschemes for other model selection problems. Second, we constructhigh-probability search paths on the space of equivalence classes with nodedegree constraints by proving a combinatorial property of DAG comparisons.Simulation studies on the proposed MCMC sampler are conducted to illustrate themain theoretical findings.

Complexity analysis of Bayesian learning of high-dimensional DAG models and their equivalence classes