Causal Identification from Counterfactual Data: Completeness and Bounding Results

Previous work establishing completeness results for counterfactual identification has been limited to the setting where the input data belongs to observational and interventional distributions (Layers 1 and 2 of Pearl's Causal Hierarchy), since it was generally presumed impossible to obtain data from counterfactual distributions, belonging to Layer 3. However, recent work (Raghavan & Bareinboim, 2025) has formally characterized a family of counterfactual distributions which can be directly estimated via experimental methods - a notion they call counterfactual realizabilty. This leaves open the question of what additional Layer 3 quantities now become identifiable, given this new access to (some) Layer 3 data. We develop the ctfIDu+ algorithm for identifying a counterfactual query from an arbitrary set of Layer 3 data, and prove that it is complete for this task. Using this, we establish the theoretical limit of which counterfactuals can be identified from physically realizable data, thus implying the fundamental limit to exact causal inference in the non-parametric setting. Finally, we derive novel analytic bounds for important non-identifiable quantities given realizable counterfactual data, that are provably tighter than the previously established benchmark. We corroborate using simulations that even if a quantity is non-identifiable, counterfactual data can be used to further tighten bounds for its range.

The Pearl Causal Hierarchy, also known as Pearl’s “ladder of causation,” organizes causal reasoning into three levels: observing associations, reasoning about interventions, and reasoning about counterfactuals. The third level concerns “what if?” questions: given what actually happened, what would likely have happened in an alternative version of the past? Such questions are central to fairness, law, and explanation, even when they are not described in causal terms. For example, legal doctrines often ask whether, all else being equal, a person would have been treated differently had their race, gender, or other protected attribute been different. Counterfactuals are difficult to validate empirically because they concern alternative histories that did not occur. This has motivated work on counterfactual identification: determining when counterfactual quantities can be inferred from data under formal causal assumptions. Until now, this literature has assumed that available data come only from the first two levels of the hierarchy: observational and interventional data. Recent advances in experimental design show that limited forms of counterfactual data can also be collected through so-called "counterfactual randomization", opening the door to a broader theory of counterfactual inference. Our work develops an algorithm that takes any collection of physically accessible input data and determines whether a target counterfactual query can be computed from that data under the stated causal assumptions. We prove that the algorithm is complete (i.e. any method that solves this task is equivalent to our algorithm), characterize the fundamental theoretical limits of counterfactual identifiability, and show that even when exact identification is impossible, counterfactual data can still reduce uncertainty about the query’s possible values.