Resolution of Simpson's paradox via the common cause principle

Simpson's paradox is an obstacle to establishing a probabilistic association between two events a₁ and a₂, given the third (lurking) random variable B. We focus on scenarios when the random variables A (which combines a₁, a₂, 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 a₁ and a₂ is to be defined via conditioning over C. This set-up generalizes the original Simpson's paradox. Now its two contradicting options simply 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 valid Simpson's paradox), the conditioning over any binary common cause C establishes the same direction of the association between a₁ and a₂ 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. For tertiary (unobserved) common causes C all three options of Simpson's paradox become possible (i.e. marginalized, conditional, and none of them), and one needs prior information on C to choose the right option.