Suppose a subscriber received no promotion and then churned. Would sending the promotion have been enough to keep them? The probability of sufficiency (PS) quantifies that counterfactual question for subscribers with the same observed history. section 8.3.1
Let X indicate whether a promotion was sent and Y whether the subscriber stayed. Write Y₁ for whether they would stay if sent the promotion. Then PS = P(Y₁ = 1 | X = 0, Y = 0). The condition selects people who actually received no promotion and churned; the probability concerns their outcome under the intervention. It is not simply the retention rate among people who received promotions. section 8.3.1
PS differs from the probability of necessity (PN). PS starts with cases where neither the cause nor the outcome occurred and asks whether adding the cause would produce the outcome. PN starts with cases where both occurred and asks whether removing the cause would prevent the outcome. Both are binary counterfactual queries used in causal attribution and uplift modeling. section 8.3.1 section 8.5
Estimating PS requires more than comparing observed groups: the outcome under promotion is unobserved for subscribers who received none. Under a monotonic, non-harmful effect of the promotion, PS can be identified as [P(Y₁ = 1) − P(Y = 1)] / P(X = 0, Y = 0), provided the denominator is nonzero. Monotonicity rules out the counterfactual possibility that sending the promotion makes a subscriber less likely to stay. The intervention probability P(Y₁ = 1) can be estimated from level-2 data such as a randomized experiment; observational data instead requires an identification strategy, such as backdoor or front-door adjustment. section 10.6.2
