Structure learning via MCMC sampling is known to be very challenging because of the enormous search space and the existence of Markov equivalent DAGs. Theoretical results on the mixing behavior are lacking. In this work, we prove the rapid mixing of a random walk Metropolis-Hastings algorithm, which reveals that the complexity of Bayesian learning of sparse equivalence classes grows only polynomially in $n$ and $p$, under some high-dimensional assumptions. A series of high-dimensional consistency results is obtained, including the strong selection consistency of an empirical Bayes model for structure learning. Our proof is based on two new results. First, we derive a general mixing time bound on finite state spaces, which can be applied to local MCMC schemes for other model selection problems. Second, we construct high-probability search paths on the space of equivalence classes with node degree constraints by proving a combinatorial property of DAG comparisons. Simulation studies on the proposed MCMC sampler are conducted to illustrate the main theoretical findings.

翻译：通过MCMC采样的结构学习因搜索空间巨大以及马尔可夫等价DAG的存在而极具挑战性，目前尚缺乏关于混合行为的理论结果。本文证明了一种随机游走Metropolis-Hastings算法的快速混合性，揭示了在高维假设下，稀疏等价类的贝叶斯学习复杂度仅随$n$和$p$呈多项式增长。我们获得了一系列高维一致性结果，包括用于结构学习的经验贝叶斯模型的强选择一致性。证明基于两个新结果：首先，推导了一般的有限状态空间混合时间界，可应用于其他模型选择问题的局部MCMC方案；其次，通过证明DAG比较的组合性质，在具有节点度约束的等价类空间上构造了高概率搜索路径。本文对所提出的MCMC采样器进行了仿真研究，以说明主要理论发现。