This paper considers the problem of testing the maximum in-degree of the Bayes net underlying an unknown probability distribution $P$ over $\{0,1\}^n$, given sample access to $P$. We show that the sample complexity of the problem is $\tilde{\Theta}(2^{n/2}/\varepsilon^2)$. Our algorithm relies on a testing-by-learning framework, previously used to obtain sample-optimal testers; in order to apply this framework, we develop new algorithms for ``near-proper'' learning of Bayes nets, and high-probability learning under $\chi^2$ divergence, which are of independent interest.

翻译：本文研究在给定分布$P$在$\{0,1\}^n$上的样本访问条件下，测试未知概率分布$P$背后贝叶斯网络的最大入度问题。我们证明该问题的样本复杂度为$\tilde{\Theta}(2^{n/2}/\varepsilon^2)$。我们的算法依赖于一种测试-学习框架，该框架先前已被用于获得样本最优测试器；为应用这一框架，我们开发了贝叶斯网络的“近真”学习新算法，以及基于$\chi^2$散度的高概率学习新算法，这些算法具有独立的研究价值。