pith. sign in

arxiv: 2605.21846 · v1 · pith:GUCZQZJMnew · submitted 2026-05-21 · 📊 stat.ME · cs.LG· stat.ML

Causal Discovery in Structural VAR Models Under Equal Noise Variance

SeyedSina Seyedi HasanAbadi , Fahimeh Arab , Erfan Nozari , AmirEmad Ghassami This is my paper

Pith reviewed 2026-05-22 05:05 UTC · model grok-4.3

classification 📊 stat.ME cs.LGstat.ML
keywords causal discoverystructural VARequal noise varianceobservational equivalencetime seriescontemporaneous effectssparsity
0
0 comments X

The pith

In structural VAR models with equal noise variances, multiple causal structures induce the same observed process and are related by orthogonal transformations plus global scale.

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

The paper studies causal discovery for linear Gaussian structural vector autoregression models when the structural noise terms share a common variance. In this time-series setting, unlike cross-sectional cases, the equal-variance condition does not produce a unique causal graph; instead, it produces an equivalence class of models that generate identical stationary observed laws. The authors define a tailored notion of observational equivalence and prove that any two models in the same class differ by an orthogonal transformation of the structural equations together with a positive global scale factor. They then introduce an equivalence-aware discrepancy measure and a sparsity-seeking algorithm, ENVAR, that returns a normalized sparse representative from within each equivalence class. A reader would care because the result supplies a principled way to perform causal discovery on multivariate time series that contain both lagged and contemporaneous effects, as arise in neuroscience applications with coarse sampling rates.

Core claim

In linear Gaussian structural VAR models under the equal noise variance assumption, the set of structural parameterizations that induce the same stationary observed process law forms an equivalence class characterized by orthogonal transformations of the structural equations together with a global positive scale. This characterization yields the observational alignment discrepancy, which compares models only after accounting for transformations that leave the observed law unchanged. Building on the theory, the ENVAR procedure searches the equivalence class for a sparse normalized structural representative.

What carries the argument

Observational equivalence class characterized by orthogonal transformations of the structural equations plus a global positive scale factor, which groups all structural VAR parameterizations that produce identical stationary observed processes.

If this is right

  • Causal discovery returns an equivalence class rather than a single graph, so downstream users must account for the remaining ambiguity.
  • The ENVAR algorithm can be run on synthetic or real multivariate time series to produce a sparse normalized representative inside the equivalence class.
  • The same observational law can arise from distinct contemporaneous causal structures provided they are related by rotation and scaling.
  • The method directly handles non-acyclic contemporaneous effects, which are realistic when sampling intervals are coarse relative to system dynamics.

Where Pith is reading between the lines

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

  • The orthogonal-plus-scale characterization may suggest analogous equivalence results for other linear time-series models that impose moment or variance constraints.
  • In fMRI or other neuroimaging pipelines, the equivalence class could be used to generate multiple candidate causal graphs that are then tested against interventional data.
  • Relaxing the equal-variance assumption would likely enlarge the equivalence class or restore point identification, offering a natural direction for sensitivity analysis.

Load-bearing premise

All structural noise terms share exactly the same variance.

What would settle it

A concrete counter-example consisting of two structural VAR parameterizations that induce the same observed process law but cannot be related by any orthogonal transformation plus global scale, or conversely an example where equal-variance models are point-identified.

Figures

Figures reproduced from arXiv: 2605.21846 by AmirEmad Ghassami, Erfan Nozari, Fahimeh Arab, SeyedSina Seyedi HasanAbadi.

Figure 1
Figure 1. Figure 1: Performance comparison of EqVarGDS, ENVAR, VARLiNGAM, and DYNOTEARS vs. [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nodal centralities of the binarized ENVAR graph during the HCP motor task. For each [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

Causal discovery from multivariate time series is challenging when causal effects may occur both across time and within the same sampling interval. This issue is especially important in applications such as neuroscience, where the sampling rate may be coarse relative to the underlying dynamics and contemporaneous effects need not form an acyclic graph. We study causal discovery in linear Gaussian structural VAR models under an equal noise variance assumption, meaning that the structural noise terms have a common variance. Unlike the DAG-based cross-sectional equal noise variance setting, the time-series setting considered here does not generally yield point identification of a unique causal graph. Instead, multiple structural VAR parameterizations can induce the same stationary observed process law. We introduce a notion of observational equivalence tailored to this setting and show that the corresponding equivalence class is characterized by orthogonal transformations of the structural equations together with a global positive scale. This characterization leads to an equivalence-aware model discrepancy, the observational alignment discrepancy, which compares structural models modulo transformations that preserve the observed law. Building on this theory, we propose ENVAR, a sparsity-based procedure that searches over the induced observational equivalence class for a sparse normalized structural representative. We evaluate the proposed methodology on synthetic structural VAR data and on an fMRI dataset.

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. This paper addresses causal discovery in linear Gaussian structural VAR models with equal noise variances. It introduces a tailored notion of observational equivalence and demonstrates that the equivalence class consists of models related by orthogonal transformations of the structural equations along with a global positive scale. Based on this, it defines an observational alignment discrepancy and develops the ENVAR procedure to identify a sparse normalized representative within the class. The methodology is assessed through experiments on synthetic structural VAR data and an fMRI dataset.

Significance. If the central theoretical result holds, the paper offers a meaningful advance in handling identifiability issues in time-series causal models. The equal variance assumption provides a realistic middle ground between full identification and complete non-identification, leading to a well-defined equivalence class that can be searched efficiently for sparse solutions. This is particularly relevant for applications with coarse sampling, such as fMRI. The direct connection to linear algebra properties of the model is a strength, and the empirical evaluations provide practical validation.

major comments (2)
  1. §3: The characterization of the equivalence class by orthogonal transformations and global scale is derived from the condition A Σ A^T = σ² I. However, the manuscript should explicitly verify that the transformation preserves the reduced-form coefficients for the lagged terms in higher-order VAR(p) models with p > 1, as this is load-bearing for the general claim.
  2. §5.2: In the fMRI application, the choice of the sparsity level and the normalization procedure within the equivalence class should be justified more clearly, as small changes could affect the recovered graph structure.
minor comments (2)
  1. Abstract: The abstract mentions 'synthetic and fMRI evaluations' but could briefly note the key performance metrics or baselines used.
  2. Introduction: Some equations use notation that is introduced later; consider moving the model definition earlier for better flow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive and constructive feedback on our manuscript. We address the major comments below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: §3: The characterization of the equivalence class by orthogonal transformations and global scale is derived from the condition A Σ A^T = σ² I. However, the manuscript should explicitly verify that the transformation preserves the reduced-form coefficients for the lagged terms in higher-order VAR(p) models with p > 1, as this is load-bearing for the general claim.

    Authors: We appreciate this important point. The equivalence transformations are defined on the structural parameters such that the observed reduced-form process is preserved. Specifically, for a VAR(p) model, applying an orthogonal transformation Q to the contemporaneous matrix A and correspondingly to the lagged coefficient matrices ensures invariance of the reduced-form coefficients. To address the referee's concern explicitly, we will add a verification subsection in §3 demonstrating algebraically that the lagged terms remain unchanged in the reduced-form representation for p > 1. This will include the relevant matrix equations. revision: yes

  2. Referee: §5.2: In the fMRI application, the choice of the sparsity level and the normalization procedure within the equivalence class should be justified more clearly, as small changes could affect the recovered graph structure.

    Authors: We agree that additional justification would improve clarity. The sparsity level was selected via a grid search minimizing the observational alignment discrepancy on held-out data, and the normalization follows directly from fixing the global scale to unity as defined in the equivalence class. In the revision, we will elaborate on this procedure in §5.2, including a brief sensitivity analysis to demonstrate robustness of the recovered graph to small perturbations in the sparsity parameter. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence class follows from linear algebra on reduced-form parameters

full rationale

The paper's central derivation establishes that, under equal noise variance, observational equivalence in structural VAR models is characterized by orthogonal transformations of the structural equations plus a global positive scale. This follows directly from the relation between structural parameters (A, lagged coefficients) and the observed stationary law (reduced-form AR matrices and innovation covariance satisfying A Sigma A^T = sigma^2 I). The characterization is obtained by algebraic manipulation of the model equations without fitting parameters to target data, without self-citation load-bearing steps, and without renaming or smuggling ansatzes. The subsequent sparsity search and observational alignment discrepancy are downstream applications of this independent mathematical result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the linear Gaussian structural VAR model class and the equal noise variance assumption; no additional free parameters or invented entities are introduced in the abstract beyond the global scale that is part of the equivalence definition.

free parameters (1)
  • global positive scale
    Part of the equivalence class that preserves the observed stationary process law; not fitted to data but arising from the characterization.
axioms (1)
  • domain assumption The data-generating process is a linear Gaussian structural VAR model in which all structural noise terms share a common variance.
    This modeling choice is invoked to define the setting and derive the equivalence class; it is stated explicitly in the abstract.

pith-pipeline@v0.9.0 · 5759 in / 1444 out tokens · 63704 ms · 2026-05-22T05:05:58.146147+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Identifiability of

    Peters, Jonas and B. Identifiability of. Biometrika , year =

    work page

  2. [2]

    Samuel , title =

    Chen, Wenyu and Drton, Mathias and Wang, Y. Samuel , title =. Biometrika , year =

    work page

  3. [3]

    Granger, Clive W. J. , title =. Econometrica , year =

    work page

  4. [4]

    , title =

    Sims, Christopher A. , title =. Econometrica , year =

    work page

  5. [5]

    New Introduction to Multiple Time Series Analysis , publisher =

    L. New Introduction to Multiple Time Series Analysis , publisher =. 2005 , doi =

    work page 2005

  6. [6]

    Causal Search in Structural Vector Autoregressive Models , booktitle =

    Moneta, Alessio and Chla. Causal Search in Structural Vector Autoregressive Models , booktitle =. 2011 , url =

    work page 2011

  7. [7]

    Estimation of a Structural Vector Autoregression Model Using Non-

    Hyv. Estimation of a Structural Vector Autoregression Model Using Non-. Journal of Machine Learning Research , year =

    work page

  8. [8]

    Causal Inference on Time Series Using Restricted Structural Equation Models , booktitle =

    Peters, Jonas and Janzing, Dominik and Sch. Causal Inference on Time Series Using Restricted Structural Equation Models , booktitle =. 2013 , url =

    work page 2013

  9. [9]

    Proceedings of the 2018

    Malinsky, Daniel and Spirtes, Peter , title =. Proceedings of the 2018. 2018 , url =

    work page 2018

  10. [10]

    Science Advances , year =

    Runge, Jakob and Nowack, Peer and Kretschmer, Marlene and Flaxman, Seth and Sejdinovic, Dino , title =. Science Advances , year =

    work page

  11. [11]

    Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence , series =

    Runge, Jakob , title =. Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence , series =. 2020 , url =

    work page 2020

  12. [12]

    Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics , series =

    Pamfil, Roxana and Sriwattanaworachai, Nisara and Desai, Shaan and Pilgerstorfer, Philip and Georgatzis, Konstantinos and Beaumont, Paul and Aragam, Bryon , title =. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics , series =. 2020 , url =

    work page 2020

  13. [13]

    and Devijver, Emilie and Gaussier, Eric , title =

    Assaad, Charles K. and Devijver, Emilie and Gaussier, Eric , title =. Journal of Artificial Intelligence Research , year =

    work page

  14. [14]

    and Harrison, Lee and Penny, Will , title =

    Friston, Karl J. and Harrison, Lee and Penny, Will , title =. NeuroImage , year =

    work page

  15. [15]

    and Miller, Karla L

    Smith, Stephen M. and Miller, Karla L. and Salimi-Khorshidi, Gholamreza and Webster, Matthew and Beckmann, Christian F. and Nichols, Thomas E. and Ramsey, Joseph D. and Woolrich, Mark W. , title =. NeuroImage , year =

    work page

  16. [16]

    and Barrett, Adam B

    Seth, Anil K. and Barrett, Adam B. and Barnett, Lionel , title =. Journal of Neuroscience , year =

    work page

  17. [17]

    and Jirsa, Viktor K

    Ritter, Petra and Schirner, Michael and McIntosh, Anthony R. and Jirsa, Viktor K. , title =. Brain Connectivity , year =

    work page

  18. [18]

    Arab, Fahimeh and Ghassami, AmirEmad and Jamalabadi, Hamidreza and Peters, Megan A. K. and Nozari, Erfan , title =. Network Neuroscience , year =

    work page

  19. [19]

    Neuroimage , volume=

    The WU-Minn human connectome project: an overview , author=. Neuroimage , volume=. 2013 , publisher=

    work page 2013

  20. [20]

    Neuroimage , volume=

    The minimal preprocessing pipelines for the Human Connectome Project , author=. Neuroimage , volume=. 2013 , publisher=

    work page 2013

  21. [21]

    Neuroimage , volume=

    Function in the human connectome: task-fMRI and individual differences in behavior , author=. Neuroimage , volume=. 2013 , publisher=

    work page 2013

  22. [22]

    Neuroimage , volume=

    Towards a consensus regarding global signal regression for resting state functional connectivity MRI , author=. Neuroimage , volume=. 2017 , publisher=

    work page 2017

  23. [23]

    Cerebral cortex , volume=

    Local-global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI , author=. Cerebral cortex , volume=. 2018 , publisher=

    work page 2018

  24. [24]

    Nature neuroscience , volume=

    Topographic organization of the human subcortex unveiled with functional connectivity gradients , author=. Nature neuroscience , volume=. 2020 , publisher=

    work page 2020

  25. [25]

    Nature Biomedical Engineering , year=

    Macroscopic resting-state brain dynamics are best described by linear models , author=. Nature Biomedical Engineering , year=

    work page

  26. [26]

    2009 , edition=

    Functional Magnetic Resonance Imaging , author=. 2009 , edition=

    work page 2009

  27. [27]

    Magnetic resonance imaging , volume=

    Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping , author=. Magnetic resonance imaging , volume=. 2003 , publisher=

    work page 2003

  28. [28]

    Nature methods , volume=

    Large-scale automated synthesis of human functional neuroimaging data , author=. Nature methods , volume=. 2011 , publisher=

    work page 2011