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 $P$, 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 $P$. 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 $P$, 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的联合概率分布并捕捉依赖关系。在一篇开创性论文中，Chickering等人（JMLR 2004）证明：给定一个定义为某贝叶斯网络边缘分布的概率分布$P$，判定是否存在一个参数有界的贝叶斯网络能够表示$P$是$\mathsf{NP}$-难问题。他们将该问题称为LEARN。本文中，我们拓展了LEARN问题的$\mathsf{NP}$-难性结果，并证明了LEARN问题的一种承诺搜索变体（即保证存在所求贝叶斯网络并要求找出该网络）同样是$\mathsf{NP}$-难的。我们通过一个关于样本复杂度的正面结果来补充上述困难性结论：该结果表明，当某个参数有界的贝叶斯网络能够表示给定分布$P$时，存在足够的样本复杂度可以恢复出一个在总变差距离意义下接近$P$的参数有界贝叶斯网络。这一结论推广了Brustle等人（EC 2020）关于度数有界情形的样本复杂度结果。