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We are interested in the problem of learning the directed acyclic graph (DAG)when data are generated from a linear structural equation model (SEM) and thecausal structure can be characterized by a polytree. Under the Gaussianpolytree models, we study sufficient conditions on the sample sizes for thewell-known Chow-Liu algorithm to exactly recover both the skeleton and theequivalence class of the polytree, which is uniquely represented by a CPDAG. Onthe other hand, necessary conditions on the required sample sizes for bothskeleton and CPDAG recovery are also derived in terms of information-theoreticlower bounds, which match the respective sufficient conditions and thereby givea sharp characterization of the difficulty of these tasks. We also considerextensions to the sub-Gaussian case, and then study the estimation of theinverse correlation matrix under such models. Our theoretical findings areillustrated by comprehensive numerical simulations, and experiments onbenchmark data also demonstrate the robustness of polytree learning when thetrue graphical structures can only be approximated by polytrees.

Learning Linear Polytree Structural Equation Models