counterfactual inference

Interventions ask what happens on average if we act; counterfactuals ask something stranger and more personal: given what actually happened to this specific unit, what would have happened to it instead, had one thing been different? It is the logic of regret and responsibility, of 'the headache would have gone away even without the pill', and it lives on the top rung of Pearl's ladder, above both observation and intervention.

In a structural causal model the answer is computed in three steps. Abduction: use the observed facts to update beliefs about the exogenous noise terms, the unit's hidden circumstances. Action: surgically alter the equation for the variable you are imagining differently. Prediction: rerun the modified model with the updated noise to read off the counterfactual outcome. Holding the noise fixed across worlds is what makes the query unit-specific rather than population-average.

Counterfactuals demand more than interventions do. Many counterfactual quantities are not identifiable from experimental data alone, because they depend on the joint behavior of two potential outcomes that are never co-observed; identification then needs functional-form assumptions such as monotonicity. This is also where causal inference meets fairness and explanation, where 'would the decision have changed had this attribute differed' is precisely a counterfactual question.