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We consider the problem of learning a Bayesian network or directed acyclicgraph (DAG) model from observational data. A number of constraint-based,score-based and hybrid algorithms have been developed for this purpose. Forconstraint-based methods, statistical consistency guarantees typically rely onthe faithfulness assumption, which has been show to be restrictive especiallyfor graphs with cycles in the skeleton. However, there is only limited work onconsistency guarantees for score-based and hybrid algorithms and it has beenunclear whether consistency guarantees can be proven under weaker conditionsthan the faithfulness assumption. In this paper, we propose the sparsestpermutation (SP) algorithm. This algorithm is based on finding the causalordering of the variables that yields the sparsest DAG. We prove that this newscore-based method is consistent under strictly weaker conditions than thefaithfulness assumption. We also demonstrate through simulations on small DAGsthat the SP algorithm compares favorably to the constraint-based PC and SGSalgorithms as well as the score-based Greedy Equivalence Search and hybridMax-Min Hill-Climbing method. In the Gaussian setting, we prove that ouralgorithm boils down to finding the permutation of the variables with sparsestCholesky decomposition for the inverse covariance matrix. Using thisconnection, we show that in the oracle setting, where the true covariancematrix is known, the SP algorithm is in fact equivalent to $\ell_0$-penalizedmaximum likelihood estimation.

Learning directed acyclic graphs based on sparsest permutations