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Causal Inference by Surrogate Experiments: z-Identifiability
Elias Bareinboim, Judea Pearl
TL;DR
When X is difficult to manipulate, the paper asks whether experiments on accessible Z can identify X’s effect. It gives a graphical criterion and complete algorithm, and establishes do-calculus completeness for z-identifiability.
Problem
The paper asks whether causal effects that are not determined by observational data and a causal graph can be computed using additional experiments on Z.
Source: §1 · p. 1Method
The paper provides a necessary-and-sufficient graphical condition and a complete algorithm that computes a formula from observational and experimental information.
Source: Conclusion · p. 7Results
The authors prove that the z-identification algorithm is complete and that do-calculus is complete for determining z-identifiability.
Source: §4 · p. 7, Conclusion · p. 7Takeaways & Limitations
In the illustrated setting, an experiment on Z suffices to identify the effect without intervening on X.
Source: §1 · p. 2The paper leaves identification with experimental data in the linear case for future work, noting that ordinary linear identifiability remains an open problem.
Source: Conclusion · p. 8
Abstract
from arXiv · showhide
We address the problem of estimating the effect of intervening on a set of variables X from experiments on a different set, Z, that is more accessible to manipulation. This problem, which we call z-identifiability, reduces to ordinary identifiability when Z = empty and, like the latter, can be given syntactic characterization using the do-calculus [Pearl, 1995; 2000]. We provide a graphical necessary and sufficient condition for z-identifiability for arbitrary sets X,Z, and Y (the outcomes). We further develop a complete algorithm for computing the causal effect of X on Y using information provided by experiments on Z. Finally, we use our results to prove completeness of do-calculus relative to z-identifiability, a result that does not follow from completeness relative to ordinary identifiability.
1 Introduction
The introduction frames z-identification as estimating effects of X using experiments on surrogate variables Z, and presents necessary-and-sufficient graphical conditions, a complete algorithm, and do-calculus completeness for the problem.
1 Introduction
- z-identification asks whether effects can be computed from observational data and a causal graph together with experiments on Z; it reduces to ordinary identification when Z is empty.Source: §1 · p. 1
- In the example, the derived expression shows that an experiment on Z can identify the effect without intervening on X.Source: §1 · p. 2
- Pearl’s earlier surrogate-variable criterion was sufficient but not shown necessary, and it was not extended to sets of variables.Source: §1 · p. 2
- In the illustrated graphs, experiments on Z identify the effect in model (a), but not in models (b) or (c).Source: Figure 1 · p. 2, §1 · p. 2
- A do-calculus expression using only interventions on Z is declarative but does not specify a derivation sequence or establish that one exists.Source: §1 · p. 2
- The paper gives a necessary-and-sufficient graphical condition for z-identification, a complete algorithm that returns formulas, and a proof that do-calculus is complete for z-identification.Source: §1 · p. 2
2 Notation and Definitions
The section defines z-identifiability as computing an effect using observational data and experiments on Z, extending ordinary identifiability. It gives a do-calculus sufficiency condition and describes how non-identifiability can be established.
2 Notation and Definitions
- One fixed level of Z suffices for identification under the assumptions in G, although several levels may be needed to obtain enough samples for desired X values.The expression is expected to give the same result regardless of the constant level at which Z is held.Source: §2 · p. 2
- The notation follows standard graph conventions, and interventions replace structural functions with constants for the intervened variables.Pa, An, and De denote parents, ancestors, and descendants; G_XZ removes incoming arrows into X and outgoing arrows from Z.Source: §2 · pp. 2–3
- Ordinary identifiability means the target effect is uniquely computable from P(V) in every model inducing G, without additional experimental information.A pair of graph-compatible models agreeing on P(V) but differing on the target effect witnesses non-identifiability.Source: §2 · p. 3
- z-identifiability asks whether an effect is uniquely computable from observational data and experiments on subsets of Z, and reduces to ordinary identifiability when Z is empty.The definition requires computability in any model inducing the causal graph.Source: §2 · p. 3
- A do-calculus derivation using only variables in Z as interventional variables is sufficient for z-identifiability.The criterion concerns reducing P(y|do(x)) to an expression with no other interventional variables.Source: §2 · p. 3
- Non-z-identifiability requires two graph-compatible models to agree on P and all available interventions on subsets of Z while differing on the target effect.This makes a non-z-identifiability proof at least as hard as a non-identifiability proof, which only requires agreement on P.Source: §2 · p. 3
3 Characterizing zID Relations
The section characterizes when experiments on Z identify a causal effect: a hedge in GZ guarantees non-identifiability, but a complete criterion must also account for Z and directed edges outside C-components. It gives necessary and sufficient conditions and illustrates why selecting suitable surrogate variables is nontrivial.
3 Characterizing zID Relations
- A hedge for Q in GZ guarantees that Q is not zID, but the hedge condition alone does not characterize all non-zID cases.The bv-graph is non-zID despite having no hedge for Q in GZ.Source: §3 · p. 4
- The p-graph is a smallest example where Z can aid identification yet the effect remains non-zID from do(Z).This case is identified as a base case for proving completeness.Source: §3 · p. 4
- Figure 3’s first-row graphs are zID from P and do(Z), whereas its second-row graphs are not; changing the graph or surrogate set can change the result.For example, adding W↔X to Fig. 3(a) makes Q non-zID for {Z} and {Z, W}.Source: Figure 3 · p. 5, §3 · p. 5
- Unlike ordinary identification, zID depends on directed edges outside a C-component, so graphs sharing an edge-subgraph hedge need not share zID status.The bv-graph illustrates this distinction.Source: §3 · pp. 4–5
- zID holds if Q is identifiable in G, or if some Z′ is intercepted from Y by X and Q is identifiable in GZ′.The criterion is both necessary and sufficient.Source: §3 · p. 5
- Selecting covariates that satisfy both theorem conditions is not immediate, and a naive search may require exponentially many tests.The next section addresses an efficient and complete algorithm for this selection problem.Source: §3 · p. 5
4 A Complete Algorithm for zID
IDz extends ordinary identification to use experiments on Z, with a sound and complete algorithm for computing causal effects. The paper also establishes that do-calculus is complete for z-identifiability and notes that its results are developed in a non-parametric setting.
4 A Complete Algorithm for zID
- IDz builds on Shpitser’s ID algorithm because its explicit use of hedges supports the completeness proof.The algorithm’s equality steps operate in interventional space, where distributions from experiments on Z are available.Source: §4 · p. 6
- The algorithm uses C-component decomposition in a divide-and-conquer strategy, exploiting shared variables when steps 3 or 4 intersect Z.This can avoid having to identify those variables and may help identify others.Source: §4 · p. 6
- Steps 3 and 4 can add subsets of Z as interventional variables and retain them through the algorithm, allowing available do(Z) distributions in the final expression.The algorithm tracks active Z-interventions through I and J.Source: §4 · pp. 6–7, Figure 4 · p. 6
- IDz is sound and complete: every returned expression is correct, and failure to identify corresponds to a hedge witness.The failure witness concerns a subproblem involving subsets of X, Y, and Z.Source: §4 · pp. 6–7
- The rules of do-calculus, together with standard probability manipulations, are complete for determining z-identifiability.IDz corresponds by construction to sequences of these operations.Source: §4 · p. 7
Conclusion
- The paper provides a graphical necessary and sufficient condition and a complete algorithm that combines observational and experimental data to compute causal effects.The algorithm addresses arbitrary sets of treatment and outcome variables using experiments on Z.Source: Conclusion · p. 7
- The results are developed in a non-parametric setting; extending experimental-data identification to the linear case remains a future direction.The paper notes that complete characterization of ordinary identifiability in the linear case is still open.Source: Conclusion · pp. 7–8