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Operationalising Relative Causal Knowledge: Backbone Identifiability from Private Reports on a Shared Outcome

This paper demonstrates that private causal reports on a shared outcome are insufficient to uniquely identify the underlying joint causal structure (backbone) due to hidden interaction degrees of freedom, but full identification becomes possible when agents communicate causally identified response functions rather than just observational summaries.

Original authors: Fabrizio Russo, Mark Somers

Published 2026-08-12

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CC BY 4.0

Original authors: Fabrizio Russo, Mark Somers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant, multi-layered mystery, but you and your friend are standing in different rooms, looking at the same crime scene through separate windows. You can see how the suspect's height affects the outcome, and your friend can see how the suspect's shoe size affects the outcome. You both write down your findings and try to combine them to understand the whole picture. This is the heart of a field called causal inference, which is basically the science of figuring out what causes what, rather than just what happens to happen together. Usually, scientists assume that if you have enough separate pieces of information, you can stitch them together to get the full truth. But what if the way you stitch them together is actually a guess? What if two different "stitched" pictures look exactly the same from your windows, but tell completely different stories about what would happen if you changed both the height and the shoe size at the same time? This paper asks a tricky question: when we share our private clues, do we actually know the secret "glue" that holds the whole story together, or are we just guessing?

This paper, titled "Operationalising Relative Causal Knowledge," dives into a specific puzzle about how different experts (or "agents") can share their knowledge about the world. The authors use a fancy mathematical idea called the "Relativity of Causal Knowledge" (RCK), which imagines a network where everyone has their own local view of reality. They want to know: if two people only know about their own specific cause-and-effect relationships, can they figure out the shared reality that connects them?

The researchers found a surprising snag. In many common situations, the answer is no. They proved that even if two agents share their perfectly accurate, private reports about how their specific causes affect a shared result, those reports might not be enough to pin down the true "backbone" of the system. Think of it like two chefs tasting a soup. One tastes the salt, the other tastes the pepper. They both report, "It's salty" and "It's peppery." But if they try to guess what the soup tastes like if you add both extra salt and extra pepper at once, they might be stuck. There could be a hidden "interaction" flavor—like a secret spice that only appears when salt and pepper mix—that neither chef can detect on their own. The paper shows that there are infinitely many different "secret spice" recipes that would make the soup taste exactly the same to each chef individually, but would taste wildly different if they both added their ingredients together. This means that without extra information, the agents are essentially guessing how their findings combine, and their guesses could lead to completely different predictions about the future.

However, the story doesn't end in a dead end. The authors show that there is a way to solve this mystery, but it requires a specific kind of honesty. If the two agents agree that their causes work independently—meaning the salt doesn't change how the pepper works, and vice versa—they can solve the puzzle. But here is the catch: they can't just share their observations (like "the soup was salty"). They have to share their causal rules (like "adding one pinch of salt increases the saltiness by exactly 0.5 units"). If they communicate these precise, cause-and-effect formulas, they can finally reconstruct the full, shared recipe.

To illustrate this, the paper uses a real-world example about education. Imagine one researcher studies how much better a teacher makes a student's future earnings, while another studies how much a better neighborhood helps. Both find positive effects. But do they know if a great teacher can cancel out a bad neighborhood, or if they work together to create something even bigger? The paper shows that just knowing the individual numbers isn't enough to answer that. You need to know the exact "function" of how the teacher helps and how the neighborhood helps, and you need to assume they don't have a hidden, magical interaction. Only then can policymakers accurately calculate how many extra years of good teaching are needed to fix a lifetime of neighborhood disadvantage.

In short, this paper proves that sharing data isn't always the same as sharing understanding. If two experts only share their partial views, they might be unknowingly building on a shaky foundation. To truly combine their knowledge, they need to agree on how their pieces fit together and share the specific rules of their game, not just the final score. It's a reminder that in science, and in life, knowing your own part of the story is great, but knowing how your part connects to everyone else's is the only way to see the whole picture clearly.

Technical Summary: Operationalising Relative Causal Knowledge

Problem Statement
The paper addresses a foundational gap in the Relativity of Causal Knowledge (RCK) framework. While RCK provides a mathematical mechanism for transporting causal knowledge between agents with different Structural Causal Models (SCMs) via a shared, interventionally consistent abstraction (a "backbone"), it presupposes the existence of this backbone. The authors investigate the prior identification problem: When do agents' private causal reports uniquely determine the shared backbone value?

Specifically, the paper focuses on a two-agent "common-effect" scenario (a collider configuration $Z_\rho \to X \leftarrow Z_\sigma$ ). Agent $\rho$ possesses private knowledge about cause $Z_\rho$ , and Agent $\sigma$ possesses private knowledge about cause $Z_\sigma$ . Both agents report causal marginals regarding a shared outcome $X$ . The core question is whether these local, single-cause reports ( $P_\rho$ and $P_\sigma$ ) are sufficient to identify the unique joint intervention kernel $Q(H | z_\rho, z_\sigma) = P(X \in H | do(Z_\rho=z_\rho, Z_\sigma=z_\sigma))$ that represents the shared backbone.

Methodology and Framework
The authors formalize the problem using the language of RCK, where causal knowledge is represented as sections on a network sheaf and cosheaf.

Projection Constraints: The local reports are modeled as projections of the joint backbone kernel $Q$ by integrating out the other agent's private cause:
$P_\rho(\cdot | z_\rho) = \int Q(\cdot | z_\rho, z_\sigma) P_{Z_\sigma}(dz_\sigma)$
$P_\sigma(\cdot | z_\sigma) = \int Q(\cdot | z_\rho, z_\sigma) P_{Z_\rho}(dz_\rho)$
Identifiability Analysis: The problem is treated as an inverse problem: given the fixed marginals $(P_\rho, P_\sigma)$ , does the set of compatible joint kernels collapse to a single element?
Perturbation Argument: To prove non-identifiability, the authors construct a family of distinct joint kernels that yield identical marginals but differ on joint interventions. This is achieved by introducing a "hidden interaction" term that vanishes upon marginalization.

Key Results

1. Generic Non-Identifiability (Theorem 1)
Under standard assumptions (existence of a compatible backbone, non-degenerate private causes with non-trivial centered directions, and a local overlap/minorisation condition), the authors prove that private causal marginals do not identify a unique backbone.

Mechanism: There exists a one-parameter family of distinct kernels $\{Q_t\}$ such that every $Q_t$ induces the exact same private reports $P_\rho$ and $P_\sigma$ . However, for $t \neq s$ , the kernels disagree on the joint interventional distribution for some $(z_\rho, z_\sigma)$ .
Interpretation: The ambiguity arises from a "hidden interaction degree of freedom." Agents can exchange truthful local summaries, but these summaries fail to specify the interaction term (e.g., whether effects are compensatory, additive, or complementary). Consequently, the edge value $\chi_\tau$ required for RCK transport is underdetermined.

2. Illustrative Witnesses
The paper provides concrete examples of this non-identifiability:

Binary Case: Analogous to a $2 \times 2$ contingency table where fixed row and column sums leave the interaction term free.
Gaussian/Linear Case: In a linear model $X = \alpha Z_\rho + \beta Z_\sigma + \gamma Z_\rho Z_\sigma + \epsilon$ , the single-cause reports identify $\alpha$ and $\beta$ but leave the interaction coefficient $\gamma$ unidentified.
Continuous Non-Gaussian: The result holds for continuous outcomes and non-Gaussian base kernels, demonstrating the obstruction is not an artifact of discrete or Gaussian assumptions.

3. Conditional Recovery (Proposition 2 and Remark 3)
The paper identifies a structural condition under which the ambiguity can be resolved: Additive Separability.

Structural Assumption: If the shared outcome satisfies $X = f_\rho(Z_\rho) + f_\sigma(Z_\sigma) + \epsilon$ , the hidden interaction term is removed.
Communication Requirement: Additivity alone is insufficient if agents only communicate observational summaries (e.g., residual variances). The authors show that even under additivity, residual summaries do not uniquely identify the backbone mean response.
Solution: Identification becomes possible only if agents communicate causally identified response functions ( $f_\rho$ and $f_\sigma$ ). Once these functions are shared, the joint mean response $m(z_\rho, z_\sigma) = f_\rho(z_\rho) + f_\sigma(z_\sigma) + E[\epsilon]$ is uniquely determined.

Significance and Claims
The paper claims to refine the operational scope of RCK by clarifying the prerequisites for causal transport:

Communication vs. Composition: The primary barrier to sharing causal knowledge in common-effect settings is a communication problem, not merely a policy-composition problem. Agents must communicate more than just marginal statistics; they must communicate the functional form of their causal responses.
Nature of Ambiguity: The identified ambiguity is causally meaningful. Compatible backbones can agree on all local observations and single-cause interventions while disagreeing fundamentally on joint interventions (e.g., whether a teacher's quality amplifies or compensates for neighborhood disadvantage).
Scope of Contribution: The authors position this work upstream of existing abstraction literature. While other works study how to transport knowledge given a shared abstraction, this paper investigates when the target of that abstraction (the backbone edge value) is identifiable from private data. It distinguishes itself from expert aggregation (which merges full models) and data fusion (which pools treatment effects), focusing specifically on the identifiability of the RCK edge object.

In the education value-added example, the paper illustrates that without communicating the specific causal response functions, a policymaker cannot determine if better teachers compensate for neighborhood disadvantage, as multiple interaction structures (compensatory, additive, or complementary) remain consistent with the same local reports.

Technical Summary: Operationalising Relative Causal Knowledge

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