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Learning a Bayesian network (BN) from data can be useful for decision-makingor discovering causal relationships. However, traditional methods often fail inmodern applications, which exhibit a larger number of observed variables thandata points. The resulting uncertainty about the underlying network as well asthe desire to incorporate prior information recommend a Bayesian approach tolearning the BN, but the highly combinatorial structure of BNs poses a strikingchallenge for inference. The current state-of-the-art methods such as orderMCMC are faster than previous methods but prevent the use of many naturalstructural priors and still have running time exponential in the maximumindegree of the true directed acyclic graph (DAG) of the BN. We here propose analternative posterior approximation based on the observation that, if weincorporate empirical conditional independence tests, we can focus on ahigh-probability DAG associated with each order of the vertices. We show thatour method allows the desired flexibility in prior specification, removestiming dependence on the maximum indegree and yields provably good posteriorapproximations; in addition, we show that it achieves superior accuracy,scalability, and sampler mixing on several datasets.

Minimal I-MAP MCMC for Scalable Structure Discovery in Causal DAG Models