The most likely common cause
This work addresses causal inference challenges in statistics and machine learning for researchers dealing with hidden variables, though it appears incremental as it builds on existing principles like maximum entropy.
The paper tackles the latent confounder problem in causal insufficiency, where a common cause exists but cannot be uniquely identified from observed joint probabilities, by applying the generalized maximum likelihood method to identify a consistent common cause, revealing non-analytic behavior in conditional probabilities during correlation transitions.
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.