Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations. In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution $\mathbb{P}$, that is defined as the marginal distribution of a Bayes net, it is $\mathsf{NP}$-hard to decide whether there is a parameter-bounded Bayes net that represents $\mathbb{P}$. They called this problem LEARN. In this work, we extend the $\mathsf{NP}$-hardness result of LEARN and prove the $\mathsf{NP}$-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net. We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution $\mathbb{P}$, that is represented by some parameter-bounded Bayes net, generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020).

翻译：贝叶斯网络在实践中被广泛用于高效表示一组随机变量上的联合概率分布并捕捉依赖关系。在Chickering等人（JMLR 2004）的奠基性论文中，他们证明了给定一个由贝叶斯网络边际分布定义的分布$\mathbb{P}$，判定是否存在一个参数有界的贝叶斯网络来表示$\mathbb{P}$是$\mathsf{NP}$难的。他们将该问题称为LEARN。在本工作中，我们扩展了LEARN问题的$\mathsf{NP}$难性结果，并证明了LEARN问题的一个承诺搜索变体也是$\mathsf{NP}$难的，该变体保证存在所讨论的贝叶斯网络并要求找出这样一个网络。我们通过一个关于样本复杂度的正面结果来补充我们的硬度结果：该结果表明，存在足够的样本复杂度可以恢复一个参数有界的贝叶斯网络，使其在总变差距离意义下接近某个由参数有界贝叶斯网络表示的给定分布$\mathbb{P}$，这推广了Brustle等人（EC 2020）关于度数有界情况的样本复杂度结果。