Counterfactual Fairness and Its Assumptions

Why Counterfactual Fairness Requires a Causal Model

A prediction can be unchanged when a sensitive attribute is removed from the model’s input list and still depend on that attribute through other variables. Conversely, a difference associated with a sensitive attribute does not by itself establish that the attribute caused the difference. To ask what a prediction would have been for the same case under a different value of the attribute, we need a model of how the variables are causally connected.

Counterfactual fairness makes this comparison at the level of a particular case, using the observed evidence about that case. It is a causal criterion, not a group error-rate measure. Its conclusion depends on the causal model, the evidence used, and which pathways from the sensitive attribute are considered acceptable.

Let \(A\) be a sensitive attribute, \(O\) the observed evidence for a case, and \(\hat{Y}\) the model prediction. The evidence is factual: it is what was observed in the actual case, including \(A=a\). For a different value \(a'\), the counterfactual prediction is evaluated using the same factual evidence to infer the case’s latent characteristics, while the causal model determines which variables change under the intervention.

The phrase “full conditional distribution” matters. For a continuous prediction, the probability of any single exact value is often zero, so equality of point probabilities would not show that two distributions are alike. Comparing the cumulative distribution functions, or probabilities over all measurable sets, avoids that problem.

How the Counterfactual Comparison Is Calculated

In an SCM, the exogenous variables \(U\) represent background factors not determined by the other variables in the model. They remain fixed when an intervention is considered. The observed evidence updates what is known about \(U\); it does not generally reveal \(U\) exactly. Consequently, a counterfactual prediction may have a distribution even when the prediction rule is deterministic for a fully specified set of variables.

Suppose a model has a mediator \(M\), which is a variable on a causal pathway from \(A\) to the prediction. Write the prediction under intervention \(a'\) as \(h(a',M_{a'}(U),U,V)\), where \(V\) represents any randomness in the prediction process. The conditional distribution is obtained by averaging over the posterior distribution of the background variables given the factual evidence:

This follows by conditioning first on \(U=u\): within each possible background state, the intervention determines the relevant mediator and prediction distribution; averaging those conditional probabilities over the posterior distribution of \(U\) gives the probability for the case as observed. If the model is deterministic once \(U\) is specified, the inner probability is an indicator: it is one when the resulting prediction lies in \(B\) and zero otherwise.

A useful sufficient condition follows directly. If changing \(A\) leaves the prediction function unchanged for every background state that has positive posterior probability, then the counterfactual prediction distributions agree. But this condition is not necessary: distinct state-by-state predictions can, in some models, produce the same overall distribution. The definition concerns distributions, not only whether a particular equation contains \(A\).

A Causal Pathway Can Change the Prediction

If \(A\) causes a mediator that the prediction uses, intervening on \(A\) may change the prediction even if \(A\) is not supplied as a direct input. For example, \(A\to M\to\hat{Y}\) is a causal pathway from the sensitive attribute to the prediction. Whether that pathway should be treated as permissible is a substantive judgment about the decision context, not a conclusion supplied by the statistical definition.

Assumptions Limit What the Result Means

A counterfactual fairness assessment requires more than a fitted prediction model. The analyst must specify a causal model: which variables cause which others, how interventions affect descendants, and how unobserved factors are represented. The model must also support the required conditional distributions. In particular, a calculation involving \(P(U\mid O,A)\) needs a justified joint distribution for those variables; independence statements must not be used to infer conditional independence that they do not imply.

The causal graph and the classification of pathways also have substantive limits. Suppose an attribute affects access to training, and training affects a measured experience variable used in a model. Treating the path through experience as permissible will produce a different fairness question from treating it as impermissible. A pathway can reflect a historical disadvantage even if it is not a direct effect of the attribute. Calling a path “permissible” does not make its consequences harmless; it records a choice about the normative standard being applied.

Counterfactual fairness is also not the same as a group-level fairness criterion. A group-level comparison summarizes rates or errors across populations. Counterfactual fairness asks whether a case’s prediction distribution would change under an intervention on \(A\), conditional on that case’s factual evidence. Either type of assessment may be informative, but neither alone establishes that a decision is appropriate in every context.

Interpreting a Fairness Conclusion

A careful conclusion states the result conditionally: under a specified SCM, for stated evidence, and under stated assumptions about the causal pathways, the prediction distributions do or do not agree across the considered interventions. It should not present the result as a model-free property of the prediction algorithm. In applied work, the causal assumptions should be explained and challenged with subject-matter experts; alternative plausible graphs may produce different conclusions.

This distinction matters in actuarial settings because variables such as prior claims, employment history, and service access may be both predictive and causally connected to sensitive attributes or historical decisions. The purpose of the analysis is not to declare a variable acceptable solely because it improves prediction, or unacceptable solely because it is associated with a sensitive attribute. It is to make the causal question explicit, test the implications of the chosen assumptions, and communicate what the result does and does not establish. This treatment is aligned with the SOA Exam PA (Predictive Analytics) syllabus.