Ignorability assumes that a causal variable and its potential outcomes are conditionally independent after conditioning on specified confounders. For exposure E, outcome I, and confounder G, this can be written as E ⊥ I^E=e | G. Conditioning on G removes the association needed to distinguish intervention effects from confounding. section 10.3.2 section 10.10
Use ignorability as an identification step:
- Represent the causal question with potential outcomes, such as the distribution of I under intervention E=e. section 10.3.2
- Condition on confounders G so that d-separation makes E independent of the relevant potential outcome. section 10.3.2
- Replace the intervention distribution with observed conditional distributions and marginalize over G, yielding the backdoor estimand ∑g P(I | E=e, G=g)P(G=g). section 10.3.2
- Apply the same substitution to effects on the treated when a SWIG establishes the required d-separation. section 10.7.4
Backdoor adjustment
Suppose E denotes exposure, I denotes in-game purchases, and G denotes a confounder. If G d-separates E from I^E=e, ignorability permits estimating the intervention distribution by summing observed outcome probabilities over G: P(I^E=e=i) = ∑g P(I=i | E=e, G=g)P(G=g). section 10.3.2
Effect on the treated
For ETT, a SWIG can justify changing the treatment condition inside a potential-outcome probability: P(A^−t=a | T=+t) = ∑c P(A=a | C=c, T=−t)P(C=c | T=+t). The change relies on C d-separating A^−t and T, followed by consistency. section 10.7.4
