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 (2013) 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 (2013) with an independent simulation study. Hence we show that NAL has a much wider applicability than originally implied in Balov (2013), and that it is competitive with EM for conditional Gaussian BNs as well.

翻译：文献中广泛研究了从完整数据中学习的巴伊西亚网络(BN)结构,然而,对于不完整数据而言,理论结果较少,而且大多与期望-最大化算法有关。Balov(2013年)提出了一个与EM具有竞争力但计算效率更高的替代方法(NAL),Beesian Network (NB) ;他证明了离散BNs的一致性和模型可识别性。在本文中,我们为NAL的一致性提供了充分的一般性条件;我们证明有条件的高斯BNs(包括作为特殊案例的离散和高斯班)的一致性和可识别性。此外,我们通过独立模拟研究确认了我们在Balov(2013年)的结果和结果。因此我们表明,NAL比原先在Balov(2013年)中所隐含的要广泛得多的适用性,而且与有条件高斯(Gaussian BNs)的EM也具有竞争力。