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arxiv: 2605.04381 · v1 · submitted 2026-05-06 · 📊 stat.ME · cs.LG· math.ST· stat.ML· stat.TH

Causal discovery under mean independence and linearity

Geert Mesters , Alvaro Ribot , Anna Seigal , Piotr Zwiernik This is my paper

Pith reviewed 2026-05-08 17:52 UTC · model grok-4.3

classification 📊 stat.ME cs.LGmath.STstat.MLstat.TH
keywords causal discoverymean independencelinear causal modelsdependent noisesource identificationacyclic graphsDirectLiMIAM
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The pith

One-sided mean independence of disturbances allows generic identification of source nodes in linear acyclic models

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

Causal discovery typically requires that noise terms be fully independent of each other. This paper replaces that with a weaker condition: each disturbance is mean-independent of the variables that cause it. Under this and linearity, the sources of the causal graph become identifiable from finite-order moment conditions. A recursive procedure can then peel off sources one by one to recover the full order. This matters when noises share volatility or other dependencies that violate strict independence but still satisfy mean independence.

Core claim

The authors define the Linear Mean-Independent Acyclic Model in which each disturbance satisfies a one-sided mean-independence restriction with respect to its parents. They prove that the finite-order consequences of these restrictions generically identify the source nodes. From any such source the graph can be reduced by removing it and its outgoing edges, and the process repeats. The proof is constructive, yielding the DirectLiMIAM algorithm that sequentially finds sources by testing mean-independence on candidate residuals.

What carries the argument

Finite-order consequences of one-sided mean-independence restrictions, used to test and identify source nodes recursively

If this is right

  • DirectLiMIAM recovers causal order by iteratively identifying and removing source nodes based on residual mean-independence tests
  • Performance exceeds LiNGAM methods in simulations where disturbances are mean-independent but otherwise dependent
  • Empirical results on oil market variables produce an ordering consistent with economic intuition, unlike independence-based methods
  • The framework shows that dependence alone does not prevent causal discovery if mean independence holds

Where Pith is reading between the lines

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

  • If mean-independence can be verified on data, the method offers a practical alternative when full independence is implausible
  • Extensions could explore whether similar finite-order conditions work for non-linear or non-Gaussian settings
  • The recursive structure suggests compatibility with other identifiability results that rely on source detection

Load-bearing premise

The one-sided mean-independence restrictions on disturbances hold and their finite-order consequences are sufficient to identify source nodes generically

What would settle it

Observing a linear acyclic system with mean-independent disturbances in which no source node satisfies the finite-order identifiability conditions, or empirical data where the recovered order is demonstrably wrong despite the restrictions

Figures

Figures reproduced from arXiv: 2605.04381 by Alvaro Ribot, Anna Seigal, Geert Mesters, Piotr Zwiernik.

Figure 1
Figure 1. Figure 1: From LiNGAM to LiMIAM. driven by ICA, which extends the logic of DirectLiNGAM [Shimizu et al., 2011] to dependent disturbances. Accordingly, the paper contributes at three levels: failure analysis, identifi￾cation theory, and methodology. For finite-moments, our viewpoint leads to an algebraic model for coupled moments under order-dependent mean independence. More broadly, it shows that structural identifi… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Linear structural equation model, (b) corresponding directed acyclic graph, and (c) order-dependent mean independent disturbances, i.e., E[ε2 | ε1] = 0 but E[ε1 | ε2] ̸= 0. causal structure is a directed acyclic graph (DAG). For notational convenience, we fix a compatible topological order and relabel variables so that parents precede children, written 1 < · · · < p. Thus, (1) Xi = X j<i βijXj + εi , i… view at source ↗
Figure 3
Figure 3. Figure 3: Exact ordering success rate across causal recovery algorithms. 5.3. Results. Figures 3-4 document the main results for the causal orderings. The order success rate and SHD show qualitatively comparable results. First, for the independent design, there are no systematic differences between the independent and mean independent algorithms. This is reassuring for the LiMIAM methods, as there do not appear to b… view at source ↗
Figure 4
Figure 4. Figure 4: Structural Hamming distance across causal recovery algorithms. where the disturbances εt = (ε1t , . . . , εpt) ′ often have an economic interpretation. This model faces a similar identification problem as the static LSEM (3), which can be resolved by imposing restrictions on A and on the distribution of the disturbances εt . An early identification approach is based on timing restrictions, which set A to b… view at source ↗
Figure 5
Figure 5. Figure 5: Estimated oil market DAGs for LiNGAM and LiMIAM. until June 2025. We include k = 24 lags in the SVAR to ensure that all serial correlation is captured and that no serial correlation remains in the errors. To this baseline oil SVAR we add a sixth variable that measures surprises in Organization of the Petroleum Exporting Countries (OPEC) announcements from high-frequency changes in oil price futures (SURPRI… view at source ↗
read the original abstract

Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces the mutual independence assumption on disturbances in linear acyclic models (as in LiNGAM) with weaker one-sided mean-independence restrictions. It claims that finite-order consequences of these restrictions generically identify source nodes, enabling recursive recovery of a compatible causal order via the constructive DirectLiMIAM algorithm. Simulations with mean-independent but dependent disturbances show outperformance over LiNGAM, and an empirical application to oil market data illustrates recovery of a realistic ordering.

Significance. If the identifiability and recursion results hold, the work is significant for causal discovery because it relaxes a strong and frequently violated assumption (full independence), addressing fragility to shared volatility or other dependence. The constructive proof leading to a sequential residual-based algorithm and the large-scale empirical example are strengths that could improve robustness in applications.

major comments (2)
  1. [§3] §3 (generic identifiability of source nodes): the argument that finite-order consequences of one-sided mean-independence suffice for generic identification of sources does not explicitly verify that these consequences (and the mean-independence restrictions themselves) are preserved in the reduced system after conditioning on or removing an identified source. Because disturbances are permitted to be dependent, removal can induce new statistical dependence in the residuals that violates the moment conditions used for identification, which is load-bearing for the recursive recovery claim.
  2. [§4] §4 (DirectLiMIAM algorithm and recursion): the sequential procedure assumes that after identifying a source the remaining variables continue to satisfy the same class of finite-order mean-independence restrictions with respect to their own disturbances and parents, but no lemma or step demonstrates this propagation under one-sided (rather than two-sided) dependence.
minor comments (2)
  1. [Abstract and §2] The abstract and §2 introduce 'finite-order consequences' without specifying the maximal order used in practice or how it is selected from data; this affects reproducibility of DirectLiMIAM.
  2. [§2] Notation for the mean-independence restriction (e.g., E[ε_i | parents] = 0 or similar) could be stated as an explicit equation rather than described in prose to avoid ambiguity with standard conditional mean independence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. The major comments raise important points regarding the completeness of our identifiability and recursion arguments. We address each comment in turn below.

read point-by-point responses

  1. Referee: [§3] §3 (generic identifiability of source nodes): the argument that finite-order consequences of one-sided mean-independence suffice for generic identification of sources does not explicitly verify that these consequences (and the mean-independence restrictions themselves) are preserved in the reduced system after conditioning on or removing an identified source. Because disturbances are permitted to be dependent, removal can induce new statistical dependence in the residuals that violates the moment conditions used for identification, which is load-bearing for the recursive recovery claim.

    Authors: We appreciate the referee highlighting this potential gap in our presentation. The manuscript establishes generic identifiability of source nodes based on the finite-order consequences of the one-sided mean-independence assumptions. However, we acknowledge that the preservation of these conditions in the reduced model after source removal is not explicitly verified, which is essential for the recursive identification. We will revise the manuscript by adding a lemma in §3 that demonstrates this preservation. Specifically, we will show that under the linear acyclic structure and the original mean-independence restrictions, the residuals in the subsystem satisfy analogous finite-order mean-independence conditions with respect to their parents, even when disturbances are dependent. This will be done by carefully tracking the induced dependencies and verifying the moment conditions hold. revision: yes

  2. Referee: [§4] §4 (DirectLiMIAM algorithm and recursion): the sequential procedure assumes that after identifying a source the remaining variables continue to satisfy the same class of finite-order mean-independence restrictions with respect to their own disturbances and parents, but no lemma or step demonstrates this propagation under one-sided (rather than two-sided) dependence.

    Authors: We agree with the referee that the DirectLiMIAM algorithm's recursive validity relies on the propagation of the assumptions. The current text assumes this without a dedicated proof step. We will update §4 to include an explicit argument or lemma establishing that after removing an identified source, the remaining variables satisfy the required mean-independence restrictions under one-sided dependence. This revision will provide the missing justification for the sequential procedure. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive identification argument is independent of inputs

full rationale

The paper's central derivation introduces the LiMIAM model with one-sided mean-independence on disturbances and claims generic identifiability of source nodes from finite-order consequences, enabling recursive order recovery via a constructive proof. This does not reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations; the abstract and described proof strategy treat the restrictions as primitive assumptions whose consequences are derived forward without circular reduction. No quoted equations or steps exhibit the target result being presupposed in the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about mean-independence of disturbances and an ad-hoc claim that finite-order consequences suffice for generic identifiability; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Disturbances satisfy one-sided mean-independence restrictions
    This is the core modeling assumption replacing full independence.
  • ad hoc to paper Finite-order consequences of mean-independence restrictions allow generic identifiability of source nodes
    This is the key technical step enabling the recursive recovery of causal order.

pith-pipeline@v0.9.0 · 5482 in / 1396 out tokens · 60862 ms · 2026-05-08T17:52:17.594104+00:00 · methodology

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