A. A. Hovhannisyan

A. Hovhannisyan, A. E. Allahverdyan

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.

A. Hovhannisyan, A. E. Allahverdyan

Simpson's paradox is an obstacle to establishing a probabilistic association between two events $a_1$ and $a_2$, given the third (lurking) random variable $B$. We focus on scenarios when the random variables $A$ (which combines $a_1$, $a_2$, and their complements) and $B$ have a common cause $C$ that need not be observed. Alternatively, we can assume that $C$ screens out $A$ from $B$. For such cases, the correct association between $a_1$ and $a_2$ is to be defined via conditioning over $C$. This setup generalizes the original Simpson's paradox: now its two contradicting options refer to two particular and different causes $C$. We show that if $B$ and $C$ are binary and $A$ is quaternary (the minimal and the most widespread situation for the Simpson's paradox), the conditioning over any binary common cause $C$ establishes the same direction of association between $a_1$ and $a_2$ as the conditioning over $B$ in the original formulation of the paradox. Thus, for the minimal common cause, one should choose the option of Simpson's paradox that assumes conditioning over $B$ and not its marginalization. The same conclusion is reached when Simpson's paradox is formulated via 3 continuous Gaussian variables: within the minimal formulation of the paradox (3 scalar continuous variables $A_1$, $A_2$, and $B$), one should choose the option with the conditioning over $B$.