Preprint identifies a limit of do-calculus for combining incomplete causal information

What it means to identify an effect

Causal inference studies what happens when variables are deliberately changed, rather than merely observed. The paper works with structural causal models: mathematical descriptions of how variables depend on their causes and independent sources of noise. A directed graph represents those relationships, including possible hidden variables.

An effect is identifiable when every model compatible with the graph and available probability distributions gives the same answer. This is a question about whether the information determines an answer uniquely—not about how accurately a finite dataset estimates it.

Do-calculus supplies rules for manipulating probability expressions involving interventions. Combined with ordinary probability operations, it is complete for several established identification problems: whenever an answer is identifiable within those settings, the rules can derive it.

The authors investigate a broader setting, which they call universal identifiability. Instead of requiring a full observational distribution or a particular structured collection of experimental distributions, it permits arbitrary collections of observational and interventional distributions over subsets of variables.

Defining the limits of the rules

The preprint, submitted to arXiv on November 6, 2026, first gives a standalone definition of completeness. Under it, an identifiable target must be transformable into an expression involving only the available distributions through a sequence of permitted rules.

Those rules include do-calculus, marginalization, conditioning, the chain rule and the complement rule, together with ordinary arithmetic rearrangements. The definition restricts derivations to transformations of terms already in the expression; it does not permit arbitrary introduction of new quantities.

The authors verify that their definition preserves previous completeness results for classical identification, general identification and conditional general identification. Their new result therefore does not overturn those narrower findings.

Three pairwise views determine a fourth

The counterexample uses four observed variables, X, Z, W and T, and one hidden variable, U. The hidden variable is a common cause of X and Z. X causes W, while Z causes T.

Only three pairwise distributions are available: P(X,Z), P(X,T) and P(Z,W). The target is P(W,T), the joint distribution of the two downstream variables. Notably, this example requires no experimental distributions. Its target is observational, which the framework permits as a case with no intervention.

The authors prove that the three available distributions uniquely determine the target. Their method uses linear algebra. They arrange P(X,Z) into a matrix and express the other available probabilities as linear equations involving unknown conditional probabilities.

A crucial feature is that these equations need not have a unique solution. The conditional distribution of W given X, for example, is not identifiable in general. Nevertheless, every solution to the relevant linear system gives the same target probability when combined with the known distribution of X and T. The intermediate details can remain ambiguous while the final answer is fixed.

Why a standard derivation cannot always work

To establish the obstruction, the authors construct a family of models in which all four observed variables are binary. They show that no single rational function—a ratio of polynomials in the available probabilities—can represent the target while remaining defined for every compatible positive model.

Their proof exploits models where the matrix used in the recovery procedure loses rank. An algebraic expression that works away from that case cannot extend to a universally defined rational formula of the required kind.

For binary variables, any finite derivation using the specified rules produces a rational expression, because its operations reduce to addition, subtraction, multiplication and division. The impossibility result therefore blocks such a universally valid derivation, despite the target being identifiable.

Scope and uncertainty

This is a theoretical preprint supported by mathematical proofs, not an empirical evaluation. The analysis assumes a given causal graph, discrete variables and a strictly positive joint distribution over observed variables. It does not measure performance with noisy estimates or finite samples.

The distinction between a single symbolic expression and a broader computational procedure is essential. The result does not say that the target cannot be computed: the paper explicitly describes recovery through solving linear equations. It establishes incompleteness under the authors’ stated derivation framework, rather than ruling out every possible algorithm.

The authors suggest that more general identification problems will require tools beyond these rule-based symbolic transformations. Their counterexample provides a specific reason: uniquely determined answers can exist even when the permitted derivations cannot express them uniformly.