Bayesian network (BN) structure learning from complete data has been extensively studied in the literature. However, fewer theoretical results are available for incomplete data, and most are related to the Expectation-Maximisation (EM) algorithm. Balov [1] proposed an alternative approach called Node-Average Likelihood (NAL) that is competitive with EM but computationally more efficient; and he proved its consistency and model identifiability for discrete BNs.

In this paper, we give general sufficient conditions for the consistency of NAL; and we prove consistency and identifiability for conditional Gaussian BNs, which include discrete and Gaussian BNs as special cases. Furthermore, we confirm our results and the results in Balov [1] with an independent simulation study. Hence we show that NAL has a much wider applicability than originally implied in Balov [1], and that it is competitive with EM for conditional Gaussian BNs as well.

贝叶斯网络 (BN) 结构从完整数据中学习已在文献中得到广泛研究。但是，对于不完整数据，可用的理论结果较少，并且大多数与期望最大化 (EM) 算法有关。Balov [1] 提出了一种称为节点平均似然（NAL）的替代方法，它与 EM 竞争但计算效率更高；他证明了离散 BN 的一致性和模型可识别性。

在本文中，我们给出了 NAL 一致性的一般充分条件；我们证明了条件高斯 BN 的一致性和可识别性，其中包括离散和高斯 BN 作为特殊情况。此外，我们通过独立的模拟研究证实了我们的结果和 Balov [1] 中的结果。因此，我们表明 NAL 具有比 Balov [1] 最初暗示的更广泛的适用性，并且它在条件高斯 BN 方面也与 EM 竞争。