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We establish finite-sample guarantees for efficient proper learning ofbounded-degree polytrees, a rich class of high-dimensional probabilitydistributions and a subclass of Bayesian networks, a widely-studied type ofgraphical model. Recently, Bhattacharyya et al. (2021) obtained finite-sampleguarantees for recovering tree-structured Bayesian networks, i.e., 1-polytrees.We extend their results by providing an efficient algorithm which learns$d$-polytrees in polynomial time and sample complexity for any bounded $d$ whenthe underlying undirected graph (skeleton) is known. We complement ouralgorithm with an information-theoretic sample complexity lower bound, showingthat the dependence on the dimension and target accuracy parameters are nearlytight.

Learning bounded-degree polytrees with known skeleton