Causal Identification from Counterfactual Data: Completeness and Bounding Results

Arvind Raghavan; Elias Bareinboim

cs.AIFeb 26, 2026

Causal Identification from Counterfactual Data: Completeness and Bounding Results

Authors: Arvind Raghavan, Elias Bareinboim

Organizations: Causal Artificial Intelligence Lab, Department of Computer Science, Columbia University

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 realizability. 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.

Figures & tables

Panel of Figure 1 — Figure 1: (a) Causal diagram for Ex. 1 (Traffic Camera); (b) Standard randomization overriding $X$ and affecting both $Z,Y$ ; (c) Counterfactual randomization of $X$ affecting $Y$ , but not $Z$ .

Method	ID Query	Input Data
IDC*	$\mathcal{L}_{3}$ query	Full $\mathcal{L}_{2}$ data
psIDC	Path-specific $\mathcal{L}_{3}$	Full $\mathcal{L}_{2}$ data
ctfID	$\mathcal{L}_{3}$ query	Subset of $\mathcal{L}_{2}$ data
$\textsc{ctfIDu}^{+}$ (ours)	$\mathcal{L}_{3}$ query	Subset of realizable $\mathcal{L}_{3}$ data

Table 1: Comparison of different algorithms for counterfactual identification and the scope of input data. Ours is complete when assuming an arbitrary set of physically realizable input data.

Figure 2: Landscape of causal identification and estimation methods. $\textsc{ctfIDu}^{+}$ is complete for $\mathcal{L}_{3}$ identification from a collection of realizable counterfactual data.

Panel of Figure 3 — Figure 3: Sugraph of a ctf-hedge

Panel of Figure 4 — Figure 4: Difference in how an intervention on $X$ affects downstream variables in $\mathcal{L}_{2}$ , $\mathcal{L}_{2.25}$ , and $\mathcal{L}_{2.5}$ .

Figure 5: $\mathcal{L}_{2.5}$ marks the theoretical limit of exact causal inference in the non-parametric setting (Thm. 4.1 ). Every layer of the PCH contains queries that may be identifiable using data from lower layers, except $\mathcal{L}_{3}\setminus\mathcal{L}_{2.5}$ .

Figure 6: Increasingly tighter partial identification bounds for NTE using data from (a) $\mathcal{L}_{1}$ , (b) $\mathcal{L}_{2}$ , and (c) $\mathcal{L}_{2.5}$ regimes.

Panel of Figure 7 — Figure 7: Example 2: partial identification bounds for NTE and NDE quantities. A density plot of values is generated by sampling from a Bayesian posterior over SCMs, given synthetic input data. The end-points of the range of values along the X-axis mark the empirically estimated range each quantity can take. Bounds are tighter when estimated using counterfactual data (blue) than interventional data (orange). Since NDE is exactly identifiable from counterfactual data, blue bounds collapse to the true value (red).

Panel of Figure 8 — Figure 8: Example 3: (Left) benefit function by unit type; (Centre) estimated bounds of population-level benefit using $\mathcal{L}_{2}$ data (orange) and sub-population level benefit using $\mathcal{L}_{2.5}$ data (blue); (Right) counterfactual strategy dominates standard interventional approach.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure 9: Algorithmic steps for counterfactual identification. Steps in the Gray-dotted box illustrate the deprecated steps of the old ctfID algorithm from prior work ( Correa et al., 2021 ) , which only allows identification from Layer 2 input data. Red boxes illustrate the new $\textsc{ctfIDu}^{+}$ algorithm (Alg. 2 ), which allows identification from Layer 3 input data. Steps 0-v are shared by both.

Panel of Figure 10 — Figure 10: Example of using $\textsc{identify}^{+}$ to identify a ctf-factor from an input ctf-factor. Each step is a recursive call to $\textsc{identify}^{+}$ .

Panel of Figure 11 — Figure 11: Example B.10 showing how the $\textsc{ctfIDu}^{+}$ algorithm (Alg. 2 ) correctly retrieves the front-door adjustment formula.

Panel of Figure 12 — Figure 12: Example B.11 involving a causal diagram (top left) and Layer 3 query (top right). The new $\textsc{ctfIDu}^{+}$ algorithm (Alg. 2 ) follows the steps shown, reaching the four ctf-factors that are both necessary and sufficient to identify, in order to identify the root query. Next, counterfactual data from an experimental regime (bottom) is used to systematically identify each of these ctf-factors.

Figure 13: Sanity check for the completeness of Alg. 2 using a randomly generated graph. The ID formula gives the correct answer for the counterfactual query in multiple random SCMs compatible with the graph. The non-identifiable query indeed elicits a counterexample of SCMs witnessing non-ID.

Figure 14: It is physically possible to draw samples from $P(y_{x},z_{x^{\prime}})$ given graph $\mathcal{G}_{1}$ using ctf-rand() , but not given $\mathcal{G}_{2}$ .

Panel of Figure 15 — Figure 15: Difference in how an intervention on $X$ affects downstream variables in $\mathcal{L}_{1}$ , $\mathcal{L}_{2}$ , $\mathcal{L}_{2.25}$ , and $\mathcal{L}_{2.5}$ .

Figure 16: Levels of inconsistency of a ctf-factor, with examples for the causal diagram shown on the right.

Figure 17: Example illustrating sections of the causal lattice generated from a graph $\mathcal{G}$ , and input data distributions as source nodes. Each subsequent node is a distribution which can be identified from all distributions pointing into it. Blue nodes are ctf-factors , each assigned an inconsistency level per Fig. 16 . $\mathcal{L}_{3}$ output nodes don’t have a valid lattice path starting from a realizable input data distribution.

Figure 18: Causal diagram and different data-collection regimes for Example 2 (Traffic Camera v2).

Panel of Figure 19 — Figure 19: Example 2 results (over 5 random underlying SCMs) showing partial identification bounds for NTE and NDE quantities. Bounds are tighter using counterfactual data (blue) than interventional data (orange). Since NDE is identifiable from counterfactual data, blue bounds are not visible as they collapse to the true value (red).

Figure 20: Causal diagram and different data-collection regimes for Example 3 (Unit Selection).

Panel of Figure 21 — Figure 21: (a) Causal graph for Example 3; (b), (c) estimated identification bounds for counterfactual query using $\mathcal{L}_{2}$ data (orange) and $\mathcal{L}_{2.5}$ data (blue) from two public datasets. Query in (c) becomes identifiable using ctf-rand() causing blue bounds to collapse to true value (in red).

Panel of Figure 23 — Figure 23: (a) Causal diagram; (b) $P(y_{x},z_{x^{\prime}})$ is realizable by joint ctf-rand() actions, but is non-ID from $\mathcal{L}_{2.25}$ or $\mathcal{L}_{2}$ data.

Panel of Figure 24 — Figure 24: Data-collection regimes corresponding to (Left) $\mathcal{A}=\emptyset$ ; (Center) $\mathcal{A}=\{\textit{rand}(X)\}$ ; (Right) $\mathcal{A}=\{\textit{ctf-rand}(X\rightarrow Y)\}$ .

Panel of Figure 25 — Figure 25: Example for regular expression under a regime (right) involving actions $\mathcal{A}=\{\textit{ctf-rand}(X\rightarrow Y),\textit{ctf-rand}(X\rightarrow W)\}$ .

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