We characterise the likelihood function computed from a Bayesian network with latent variables as root nodes. We show that the marginal distribution over the remaining, manifest, variables also factorises as a Bayesian network, which we call empirical. A dataset of observations of the manifest variables allows us to quantify the parameters of the empirical Bayesian net. We prove that (i) the likelihood of such a dataset from the original Bayesian network is dominated by the global maximum of the likelihood from the empirical one; and that (ii) such a maximum is attained if and only if the parameters of the Bayesian network are consistent with those of the empirical model.

翻译：我们刻画了以潜变量作为根节点的贝叶斯网络所计算的似然函数。研究表明，其余显式变量的边缘分布同样可以分解为一个贝叶斯网络，我们将其称为经验贝叶斯网络。显式变量的观测数据集使我们能够量化经验贝叶斯网络的参数。我们证明：(i) 原始贝叶斯网络对该数据集的似然函数受限于经验贝叶斯网络的全局最大似然值；(ii) 当且仅当贝叶斯网络的参数与经验模型的参数一致时，该最大值方可达到。