Causal Identification from Counterfactual Data: Completeness and Bounding Results

TL;DR

This paper introduces a complete algorithm for identifying counterfactual queries from Layer 3 distributions, establishing theoretical limits of causal inference and deriving bounds for non-identifiable counterfactuals using realizable data.

Contribution

It develops the CTFIDU+ algorithm for counterfactual identification from Layer 3 data and proves its completeness, expanding causal inference capabilities.

Findings

CTFIDU+ algorithm is complete for identifying counterfactuals from Layer 3 data.

Theoretical limits of counterfactual identification are established.

Bounds for non-identifiable counterfactuals are derived and validated through simulations.

Abstract

Previous work establishing completeness results for counterfactual identification has been circumscribed to the setting where the input data belongs to observational or interventional distributions (Layers 1 and 2 of Pearl's Causal Hierarchy), since it was generally presumed impossible to obtain data from counterfactual distributions, which belong 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 counterfactual quantities now become identifiable, given this new access to (some) Layer 3 data. To answer this question, we develop the CTFIDU+ algorithm for identifying counterfactual queries from an arbitrary set of Layer 3 distributions, and prove that it is complete for this task. Building on this, we establish the theoretical limit of which counterfactuals can be identified from physically realizable distributions, thus implying the fundamental limit to exact causal inference in the non-parametric setting. Finally, given the impossibility of identifying certain critical types of counterfactuals, we derive novel analytic bounds for such quantities using realizable counterfactual data, and corroborate using simulations that counterfactual data helps tighten the bounds for non-identifiable quantities in practice.