A churn-risk score predicts who may leave; it does not explain whether a promotion kept a particular subscriber. Probabilities of causation ask counterfactual “why” questions about a binary cause X and outcome Y. Let X=1 mean a promotion was sent and Y=1 mean the subscriber stayed. Write Y₁ and Y₀ for whether that subscriber would stay with and without the promotion. section 8.3.1 section 8.5
Three related queries answer different questions:
- Probability of necessity (PN): P(Y₀=0 | X=1, Y=1). Among subscribers who received a promotion and stayed, how likely is it that they would have left without it? section 8.3.1
- Probability of sufficiency (PS): P(Y₁=1 | X=0, Y=0). Among subscribers who received no promotion and left, how likely is it that a promotion would have kept them? section 8.3.1 section 10.6.2
- Probability of necessity and sufficiency (PNS): P(Y₁=1, Y₀=0). How likely is it that the promotion changes a subscriber’s outcome from leaving to staying? section 10.6.2
These are not interchangeable with a churn prediction or an average treatment effect: PN and PS condition on different observed groups, while PNS asks about both potential outcomes together. Enablement and disablement are also named probabilities of causation for binary causes and outcomes. section 8.3.1 section 10.6.2
Identification needs assumptions. If sending the promotion cannot make anyone less likely to stay, monotonicity eliminates counterfactual possibilities. Provided their denominators are positive, PN = [P(Y=1) − P(Y₀=1)] / P(X=1,Y=1), PS = [P(Y₁=1) − P(Y=1)] / P(X=0,Y=0), and PNS = P(Y₁=1) − P(Y₀=1). The intervention probabilities can come from randomized data or, with a valid identification strategy, observational data. Without full identification, assumption-specific bounds may still constrain a probability of causation. section 10.6.2 section 10.9
Attributing a network crash
Suppose a person accessed a faulty server before a network crash. Sufficiency asks whether that access could have triggered the crash; necessity asks whether the crash would have been avoided without it. Neither probability alone settles blame: an attribution measure may also consider how unusual the access was. section 8.3.2
