The common cause principle for two random variables $A$ and $B$ is examined in the case of causal insufficiency, when their common cause $C$ is known to exist, but only the joint probability of $A$ and $B$ is observed. As a result, $C$ cannot be uniquely identified (the latent confounder problem). We show that the generalized maximum likelihood method can be applied to this situation and allows identification of $C$ that is consistent with the common cause principle. It closely relates to the maximum entropy principle. Investigation of the two binary symmetric variables reveals a non-analytic behavior of conditional probabilities reminiscent of a second-order phase transition. This occurs during the transition from correlation to anti-correlation in the observed probability distribution. The relation between the generalized likelihood approach and alternative methods, such as predictive likelihood and the minimum common cause entropy, is discussed. The consideration of the common cause for three observed variables (and one hidden cause) uncovers causal structures that defy representation through directed acyclic graphs with the Markov condition.

翻译：针对两个随机变量$A$和$B$的共同原因原理，本研究考察了因果不充分的情形：已知二者的共同原因$C$存在，但仅能观测到$A$和$B$的联合概率分布。在此条件下，$C$无法被唯一确定（潜在混杂因子问题）。我们证明广义最大似然方法可适用于此场景，并能识别出符合共同原因原理的$C$。该方法与最大熵原理密切相关。对两个二元对称变量的研究表明，条件概率存在非解析行为，其特性类似于二阶相变。这种现象出现在观测概率分布从相关到反相关的转变过程中。本文探讨了广义似然方法与预测似然、最小共同原因熵等其他方法的关联。通过对三个观测变量（及一个隐藏原因）的共同原因分析，发现了无法通过满足马尔可夫条件的有向无环图表示的因果结构。