Causal Additive Models with Unobserved Causal Paths and Backdoor Paths

Shohei Shimizu; Takashi Nicholas Maeda; Thong Pham

arxiv: 2502.07646 · v3 · pith:Z7TE26CEnew · submitted 2025-02-11 · 💻 cs.LG · stat.ME· stat.ML

Causal Additive Models with Unobserved Causal Paths and Backdoor Paths

Thong Pham , Takashi Nicholas Maeda , Shohei Shimizu This is my paper

Pith reviewed 2026-05-25 08:17 UTC · model grok-4.3

classification 💻 cs.LG stat.MEstat.ML

keywords causal discoverycausal additive modelshidden variablesbackdoor pathsregression residualsconditional independencesearch algorithm

0 comments

The pith

Causal directions in additive models can be identified even with unobserved backdoor and causal paths under new regression conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that causal directions remain identifiable in causal additive models even when unobserved backdoor or causal paths exist between variables. This matters because many real-world causal systems involve hidden variables that block standard identification methods. The key is new characterizations of regression sets that correctly detect independence of residuals and conditional independencies among observed variables. These lead to a search algorithm proven sound and complete for recovering the causal structure.

Core claim

We establish sufficient conditions under which causal directions can be identified in many such cases. These conditions rely on new characterizations of regression sets to determine independence among regression residuals and conditional independencies among observed variables. Building on these results, we introduce a search algorithm that incorporates these innovations and prove its soundness and completeness.

What carries the argument

New characterizations of regression sets that determine independence among regression residuals and conditional independencies among observed variables, even with unobserved paths.

If this is right

Many previously unidentifiable causal relationships become identifiable under the stated conditions.
A search algorithm exists that is sound and complete for recovering the structure.
The method applies to both unobserved backdoor paths and unobserved causal paths.
Empirical performance is competitive with state-of-the-art causal discovery methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The regression characterizations might extend to non-additive causal models if similar independence properties can be established.
Practical use could improve structure learning in domains with latent variables such as gene regulatory networks.
One could validate the conditions by generating data with controlled hidden paths and measuring how often residual independence matches the predictions.

Load-bearing premise

The new characterizations of regression sets correctly determine independence among regression residuals and conditional independencies among observed variables even when unobserved backdoor or causal paths exist.

What would settle it

A dataset generated from a known causal additive model with an unobserved path where the claimed residual independence fails to hold, causing the search algorithm to output an incorrect or incomplete causal structure.

Figures

Figures reproduced from arXiv: 2502.07646 by Shohei Shimizu, Takashi Nicholas Maeda, Thong Pham.

**Figure 3.** Figure 3: Performance in BA random graphs with Gaussian [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Results on sociology data. a: ground truth based on domain knowledge [Duncan et al., 1972]. A bidirected edge indicates that the relation is not modeled. A dashed directed edge represents an ancestor relationship. b: result by CAM-UV. A solid directed edge denotes a visible parent–child relationship (adjacency). An empty edge denotes a visible non-edge (non-adjacency). A bidirected edge denotes an invisi… view at source ↗

read the original abstract

Causal additive models provide a tractable yet expressive framework for causal discovery in the presence of hidden variables. When unobserved backdoor or causal paths exist between two variables, their causal relationship is often unidentifiable under existing theories. We establish sufficient conditions under which causal directions can be identified in many such cases. These conditions rely on new characterizations of regression sets to determine independence among regression residuals and conditional independencies among observed variables. Building on these results, we introduce a search algorithm that incorporates these innovations and prove its soundness and completeness. Empirical evaluations demonstrate its competitive performance against state-of-the-art methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives new regression-set characterizations that support identifying causal directions in additive models even with hidden backdoor or causal paths, plus a sound and complete search algorithm, but the characterizations are the load-bearing piece that needs verification.

read the letter

The main thing is that the authors extend causal additive models to cover more cases with unobserved paths by defining new characterizations of regression sets. These are meant to recover independence among residuals and conditional independencies among observed variables, which then lets them state sufficient conditions for direction identification. They follow this with a search algorithm and prove it sound and complete, plus some empirical checks against other methods. That is the concrete advance over prior work on additive models that stopped at identifiable cases without hidden paths. The empirical part shows the algorithm is at least competitive, which is useful for seeing whether the theory translates to data. The characterizations themselves are the part that does the real lifting here. The soft spot is exactly whether those characterizations hold when backdoor or causal paths are hidden. The claims rest on them correctly determining the needed independencies in the stated generality. If a characterization misses some graph structure or requires extra conditions not spelled out, the identification results and the completeness proof would not apply as broadly. The paper asserts the proofs, so the derivations will show whether the conditions are tight or loose. This is for people already working on causal discovery with latent variables and additive assumptions. Someone who knows the LiNGAM or CAM literature would follow the technical steps and could test the algorithm on their own graphs. It deserves a serious referee because it targets a standing limitation with a specific technical fix and supplies both the theory and the experiments. I would send it out for review.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to establish sufficient conditions for identifying causal directions in causal additive models even when unobserved backdoor or causal paths exist between variables. These conditions rely on new characterizations of regression sets to determine independence among regression residuals and conditional independencies among observed variables. It introduces a search algorithm incorporating these results, proves its soundness and completeness, and shows competitive empirical performance against state-of-the-art methods.

Significance. If the characterizations hold, the work would meaningfully extend causal discovery to graphs with hidden paths that prior theories leave unidentifiable. The manuscript ships explicit soundness and completeness proofs for the algorithm, which is a clear strength, along with an empirical evaluation.

major comments (1)

[Abstract] Abstract: The claim that the new characterizations of regression sets correctly determine independence among regression residuals and conditional independencies among observed variables even when unobserved backdoor or causal paths exist is the sole basis for the sufficient conditions and for the soundness/completeness proof of the search algorithm. The abstract supplies no counter-example checks or proof sketches, leaving this load-bearing step unverified.

minor comments (1)

The abstract could include a short illustrative example of a regression set characterization to help readers assess the scope of the new conditions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for identifying the central role of the regression-set characterizations. We address the major comment below and propose a targeted revision to the abstract.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that the new characterizations of regression sets correctly determine independence among regression residuals and conditional independencies among observed variables even when unobserved backdoor or causal paths exist is the sole basis for the sufficient conditions and for the soundness/completeness proof of the search algorithm. The abstract supplies no counter-example checks or proof sketches, leaving this load-bearing step unverified.

Authors: We agree that the abstract, as a concise summary, does not itself contain proof sketches or counter-example verification; those appear in the body of the manuscript. Section 3 derives the new regression-set characterizations, proves that they correctly identify residual independence and the relevant conditional independencies even in the presence of unobserved backdoor and causal paths, and supplies the supporting lemmas. Section 4 then uses these results to establish soundness and completeness of the search algorithm. While we believe the current abstract accurately reflects the paper’s contributions, we acknowledge that a brief additional clause could make the load-bearing step more visible to readers who stop at the abstract. We will therefore revise the abstract to include one sentence noting that the characterizations are formally proven in Section 3 and that they extend existing additive-model theory to graphs containing hidden paths. revision: yes

Circularity Check

0 steps flagged

No circularity: new characterizations and algorithm are independently derived

full rationale

The paper introduces novel characterizations of regression sets for determining residual independence and conditional independencies under hidden paths, then uses them to establish sufficient conditions for causal direction identification and to prove soundness/completeness of a new search algorithm. These steps are presented as original contributions rather than reductions to prior fits, self-citations, or definitional equivalences. No quoted equations or claims in the provided text show a result being equivalent to its inputs by construction, and the central claims retain independent mathematical content beyond any referenced prior CAM literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work rests on the standard assumptions of causal additive models and the validity of the new regression characterizations.

axioms (1)

domain assumption Causal additive model framework with additive noise
The paper builds directly on causal additive models as the base framework for handling hidden variables.

pith-pipeline@v0.9.0 · 5627 in / 1228 out tokens · 32316 ms · 2026-05-25T08:17:11.868027+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

new characterizations of regression sets to determine independence among regression residuals and conditional independencies among observed variables... CAM-UV-X algorithm... soundness and completeness
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

visible parent... invisible pairs... UBP/UCP... bow pattern

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Adams, N

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C. Glymour, K. Zhang, and P. Spirtes. Review of causal discovery methods based on graphical models. Frontiers in genetics, 10: 0 524, 2019

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Gretton, K

A. Gretton, K. Fukumizu, C. Teo, L. Song, B. Sch\" o lkopf, and A. Smola. A kernel statistical test of independence. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems, volume 20. Curran Associates, Inc., 2007. URL https://proceedings.neurips.cc/paper_files/paper/2007/file/d5cfead94f5350c12c322b5b6...

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P. O. Hoyer, S. Shimizu, A. Kerminen, and M. Palviainen. Estimation of causal effects using linear non- Gaussian causal models with hidden variables. International Journal of Approximate Reasoning, 49 0 (2): 0 362--378, 2008

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P. O. Hoyer, D. Janzing, J. Mooij, J. Peters, and B. Sch\" o lkopf. Nonlinear causal discovery with additive noise models. In Advances in Neural Information Processing Systems 21 , pages 689--696. Curran Associates Inc., 2009

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T. N. Maeda and S. Shimizu. RCD : Repetitive causal discovery of linear non- G aussian acyclic models with latent confounders. In Proc. 23rd International Conference on Artificial Intelligence and Statistics (AISTATS2010), volume 108 of Proceedings of Machine Learning Research, pages 735--745. PMLR, 26--28 Aug 2020

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