Dependencies on the relative frequency of a state in the domain are common when modelling probabilistic dependencies on relational data. For instance, the likelihood of a school closure during an epidemic might depend on the proportion of infected pupils exceeding a threshold. Often, rather than depending on discrete thresholds, dependencies are continuous: for instance, the likelihood of any one mosquito bite transmitting an illness depends on the proportion of carrier mosquitoes. Current approaches usually only consider probabilities over possible worlds rather than over domain elements themselves. An exception are the recently introduced Lifted Bayesian Networks for Conditional Probability Logic, which express discrete dependencies on probabilistic data. We introduce functional lifted Bayesian networks, a formalism that explicitly incorporates continuous dependencies on relative frequencies into statistical relational artificial intelligence. and compare and contrast them with ifted Bayesian Networks for Conditional Probability Logic. Incorporating relative frequencies is not only beneficial to modelling; it also provides a more rigorous approach to learning problems where training and test or application domains have different sizes. To this end, we provide a representation of the asymptotic probability distributions induced by functional lifted Bayesian networks on domains of increasing sizes. Since that representation has well-understood scaling behaviour across domain sizes, it can be used to estimate parameters for a large domain consistently from randomly sampled subpopulations. Furthermore, we show that in parametric families of FLBN, convergence is uniform in the parameters, which ensures a meaningful dependence of the asymptotic probabilities on the parameters of the model.

翻译：在关系数据上建模概率依赖时，通常需要考虑领域中状态相对频率的依赖关系。例如，疫情期间学校关闭的可能性可能取决于感染学生比例是否超过阈值。不同于离散阈值，依赖关系往往是连续的：例如，蚊子叮咬传播疾病的可能性取决于携带病毒的蚊子占比。现有方法通常仅考虑可能世界上的概率，而非领域元素本身的概率。最近提出的条件概率逻辑提升贝叶斯网络是一个例外，它表达了对概率数据的离散依赖关系。我们提出了函数化提升贝叶斯网络，这是一种将相对频率的连续依赖显式纳入统计关系人工智能的形式化方法，并将其与条件概率逻辑提升贝叶斯网络进行了对比。引入相对频率不仅有利于建模，还为训练集与测试集（或应用领域）规模不同的学习问题提供了更严谨的解决方案。为此，我们给出了由函数化提升贝叶斯网络在递增规模领域上导出的渐近概率分布表示。由于该表示在领域规模上具有清晰理解的缩放行为，它可用于从随机抽样子总体一致地估计大领域参数。此外，我们证明在FLBN的参数族中，收敛性在参数上是一致的，这确保了渐近概率对模型参数的有意义依赖关系。