Consistency Guarantees for Permutation-Based Causal Inference Algorithms

Bayesian networks, or directed acyclic graph (DAG) models, are widely used to represent complex causal systems. Since the basic task of learning a Bayesian network from data is NP-hard, a standard approach is greedy search over the space of DAGs or Markov equivalent DAGs. Since the space of DAGs on $p$ nodes and the associated space of Markov equivalence classes are both much larger than the space of permutations, it is desirable to consider permutation-based searches. We here provide the first consistency guarantees, both uniform and high-dimensional, of a permutation-based greedy search. Geometrically, this search corresponds to a simplex-type algorithm on a sub-polytope of the permutohedron, the DAG associahedron. Every vertex in this polytope is associated with a DAG, and hence with a collection of permutations that are consistent with the DAG ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated DAGs. We show based on simulations that this permutation search is competitive with standard approaches.