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arxiv: 2501.19311 · v4 · submitted 2025-01-31 · 📊 stat.ME

The Case for Time in Causal DAGs

Alexander G. Reisach , Alberto Su\'arez , Sebastian Weichwald , Antoine Chambaz This is my paper

Pith reviewed 2026-05-23 04:23 UTC · model grok-4.3

classification 📊 stat.ME
keywords causal DAGstime orderacyclicitytemporal qualificationcomposite variablescausalitydirected graphs
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The pith

Causal DAGs without time are ambiguous and block clear justification of no cycles.

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

The paper argues that standard causal directed acyclic graphs omit any reference to time, which leaves open multiple possible orderings of the variables and therefore multiple possible interpretations of which edges represent causation. Because the authors take it as given that a cause must occur before its effect, the direction and existence of a causal link depend on the chosen time order. Without that order the graph cannot unambiguously encode the claimed causal structure. The same omission also prevents a principled defense of the acyclicity assumption, since cycles become possible or impossible only once time is fixed. The authors therefore introduce composite causal variables that bundle values observed at one or more explicit time points so that temporal qualification is built into the representation itself.

Core claim

Assuming that causes precede effects, causal relationships are relative to the time order, and causal DAGs require temporal qualification. Nontemporal causal DAGs are therefore ambiguous and obstruct justification of the acyclicity assumption. The paper proposes a formalization via composite causal variables that refer to quantities at one or multiple time points and notes that the justification for acyclicity itself changes according to whether the time order permits cycles.

What carries the argument

Composite causal variables that refer to quantities at one or multiple time points, supplying the temporal qualification that turns an otherwise ambiguous graph into a determinate causal claim.

If this is right

  • The justification offered for acyclicity must be stated separately for time orders that allow cycles and those that do not.
  • Any causal interpretation attached to a DAG changes once the time points attached to its variables are made explicit.
  • The range of data sets and domains to which DAG causal models can be applied is restricted by the availability of a consistent time order.

Where Pith is reading between the lines

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

  • Causal discovery procedures that output graphs would need to record or infer the associated time order to produce unambiguous results.
  • Settings with irregularly sampled or asynchronous observations may require a more flexible version of the composite-variable construction.

Load-bearing premise

That every cause must precede its effect in time.

What would settle it

A concrete nontemporal causal DAG together with a demonstration that its causal interpretation is unambiguous and that its acyclicity can be justified without any reference to time order.

Figures

Figures reproduced from arXiv: 2501.19311 by Alberto Su\'arez, Alexander G. Reisach, Antoine Chambaz, Sebastian Weichwald.

Figure 1
Figure 1. Figure 1: Timeline of instantiated aspirin (◦) and headache (◦) variables. It follows from the causal order that the DAG models a time order of aspirin before headache, as shown in Figure 1a, but it is unclear whether it also models the inverse order of headache before aspirin, as shown in Figure 1b. If the DAG is taken to describe only aspirin before headache, then the absence of an effect of H on A follows purely … view at source ↗
Figure 2
Figure 2. Figure 2: Composite aspirin and headache variables referring to quantities at multiple time points. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Four realizations with their respective timelines along which causal effects unfold. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Atomic causal DAG. 3.2 A Tale of Two Acyclicities A causal cycle between two variables requires that at least one of them refers to quantities at multiple time points, as established in Remark 1. This has important implications for the justification of the acyclicity assumption (see Definition A.3 for a formal definition of acyclicity). We partition the acyclicity assumption into two separate and individua… view at source ↗
Figure 8
Figure 8. Figure 8: Definition of composite variables and illustrative timeline. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The atomic DAG and cyclic causal directed graph for the variable composition shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: time 0 ◦ YTy1 7 ◦ XTx 10 ◦ YTy2 (a) Timeline and composition of the unrolled causal variables. YTy1 XTx YTy2 (b) Unrolled time-acyclic causal DAG [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: An illustration of Example 5. The temporal aggregation of the atomic causal DAG leads to a Bayesian network between composite variables that contains an edge pointing backward in time. The temporal aggregation of the atomic causal DAG in Figure 12a leads to a pairwise causal dependence between XTx and YTy and between YTy and ZTz , but not between XTx and ZTz . Applying the orientation rules in Meek 1995a … view at source ↗
read the original abstract

We make the case for incorporating a notion of time into causal directed acyclic graphs (DAGs). We demonstrate that nontemporal causal DAGs are ambiguous and obstruct justification of the acyclicity assumption. Assuming that causes precede effects, causal relationships are relative to the time order, and causal DAGs require temporal qualification. We propose a formalization via composite causal variables that refer to quantities at one or multiple time points. We emphasize that the acyclicity assumption requires different justifications depending on whether the time order allows cycles. We conclude by discussing implications for the interpretation and applicability of DAGs as causal models.

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

1 major / 1 minor

Summary. The paper claims that nontemporal causal DAGs are ambiguous and obstruct justification of the acyclicity assumption. Assuming that causes precede effects, it argues that causal relationships are relative to time order and thus require temporal qualification. It proposes formalization via composite causal variables referring to quantities at one or multiple time points, emphasizes that acyclicity justifications differ by time order, and discusses implications for DAG interpretation and applicability.

Significance. If the core argument and formalization hold, the work would identify a foundational ambiguity in standard causal graphical models used in statistics, potentially leading to revised modeling practices that explicitly incorporate time to strengthen acyclicity justifications. The composite-variable approach offers a concrete mechanism that could be extended in applied causal inference.

major comments (1)
  1. [Abstract] Abstract (and the central argument): the claim that nontemporal DAGs are ambiguous and that acyclicity requires temporal justification rests entirely on the premise that causes precede effects. No derivation, defense, or discussion of exceptions (simultaneous causation, feedback loops without strict temporal precedence, or relativistic/quantum settings) is indicated, so the necessity of temporal qualification for causal DAGs in general does not follow.
minor comments (1)
  1. The abstract would benefit from a short statement positioning the composite-variable proposal relative to prior temporal extensions of DAGs in the causal inference literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed comment. We address it point by point below, clarifying the intended scope of the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the central argument): the claim that nontemporal DAGs are ambiguous and that acyclicity requires temporal justification rests entirely on the premise that causes precede effects. No derivation, defense, or discussion of exceptions (simultaneous causation, feedback loops without strict temporal precedence, or relativistic/quantum settings) is indicated, so the necessity of temporal qualification for causal DAGs in general does not follow.

    Authors: The manuscript explicitly conditions its argument on the assumption that causes precede effects, as stated in the abstract and repeated in the introduction and body. The claims about ambiguity in nontemporal DAGs and the need for temporal qualification to justify acyclicity are developed strictly within this standard framework common to statistical causal inference. The paper does not derive or defend the temporal-precedence assumption itself, nor does it claim necessity outside settings where the assumption holds; exceptions such as simultaneous causation or non-classical physical settings lie beyond the paper's scope. We will revise the abstract and introduction to state this delimitation more explicitly, noting that the argument applies under the stated assumption without extension to general cases. revision: partial

Circularity Check

0 steps flagged

No circularity; argument is self-contained conceptual case resting on explicit premise

full rationale

The paper advances a conceptual argument that nontemporal DAGs are ambiguous under the stated assumption that causes precede effects. No equations, fitted parameters, predictions, or self-citations are invoked in a load-bearing way that reduces the central claim to its own inputs by construction. The premise is declared upfront rather than smuggled in via definition or prior self-work, and the formalization via composite variables is presented as a proposal rather than a derived necessity. This matches the default expectation of a non-circular paper whose reasoning chain does not collapse into self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption of temporal precedence of causes and introduces composite causal variables as a new construct without independent evidence provided in the abstract.

axioms (1)
  • domain assumption Causes precede effects
    Explicitly invoked in the abstract to establish that causal relationships are relative to time order.
invented entities (1)
  • composite causal variables no independent evidence
    purpose: To refer to quantities at one or multiple time points for formalizing temporal qualification in causal DAGs
    Proposed in the abstract as the mechanism for incorporating time; no independent evidence or falsifiable handle is mentioned.

pith-pipeline@v0.9.0 · 5626 in / 1082 out tokens · 34379 ms · 2026-05-23T04:23:07.356363+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

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    cs.AI 2026-04 unverdicted novelty 6.0

    Collective agency arises when a group's joint actions are faithfully captured by a simpler causal model of unified rational behavior.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aalen, Kjetil Røysland, Jon M

    Odd O. Aalen, Kjetil Røysland, Jon M. Gran, Roger Kouyos, and Tanja Lange. Can we believe the DAGs? A comment on the relationship between causal DAGs and mechanisms. In:Statistical Methods in Medical Research25.5 (2016), pp. 2294–2314 (cit. on p. 5)

    work page 2016

  2. [2]

    Anand, Adele H

    Tara V. Anand, Adele H. Ribeiro, Jin Tian, and Elias Bareinboim. Causal Effect Identification in Cluster DAGs. In:Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 37. 10. 2023, pp. 12172–12179 (cit. on pp. 6, 9)

    work page 2023

  3. [3]

    Angrist and Jörn-Steffen Pischke

    Joshua D. Angrist and Jörn-Steffen Pischke. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton University Press, 2009 (cit. on p. 12)

    work page 2009

  4. [4]

    Causal Discovery between time series

    Charles Assaad. Causal Discovery between time series. Theses. Université Grenoble Alpes, July 2021 (cit. on p. 6)

    work page 2021

  5. [5]

    Stephan Bongers, Patrick Forré, Jonas Peters, and Joris M. Mooij. Foundations of Structural Causal Models with Cycles and Latent Variables. In:The Annals of Statistics49.5 (2021), pp. 2885–2915 (cit. on pp. 6, 8)

    work page 2021

  6. [6]

    Kording, Karen Sachs, Alexandre Drouin, and Dhanya Sridhar

    Philippe Brouillard, Chandler Squires, Jonas Wahl, Konrad P. Kording, Karen Sachs, Alexandre Drouin, and Dhanya Sridhar. The Landscape of Causal Discovery Data: Grounding Causal Discovery in Real-World Applications. In:Causal Learning and Reasoning (CLeaR). 2025 (cit. on pp. 9, 12)

    work page 2025

  7. [7]

    Invariance, causality and robustness

    Peter Bühlmann. Invariance, causality and robustness. In:Statistical Science35.3 (2020), pp. 404–426 (cit. on p. 12)

    work page 2020

  8. [8]

    Philip A. Dawid. Beware of the DAG! In:Causality: Abjectives and Assessment. PMLR. 2010, pp. 59–86 (cit. on pp. 5, 22)

    work page 2010

  9. [9]

    A model for reasoning about persistence and causation

    Thomas Dean and Keiji Kanazawa. A model for reasoning about persistence and causation. In: Computational intelligence 5.2 (1989), pp. 142–150 (cit. on p. 6)

    work page 1989

  10. [10]

    The Case for Evaluating Causal Models Using Interventional Measures and Empirical Data

    Amanda Gentzel, Dan Garant, and David Jensen. The Case for Evaluating Causal Models Using Interventional Measures and Empirical Data. In:Advances in Neural Information Processing Systems (NeurIPS). Vol. 32. 2019 (cit. on p. 12)

    work page 2019

  11. [11]

    Clive W. J. Granger. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. In:Econometrica 37.3 (Aug. 1969), pp. 424–438 (cit. on p. 2)

    work page 1969

  12. [12]

    Overthrowing the Tyranny of Null Hypotheses Hidden in Causal Diagrams

    Sander Greenland. Overthrowing the Tyranny of Null Hypotheses Hidden in Causal Diagrams. In: Heuristics, Probability and Causality: A Tribute to Judea Pearl(2010), pp. 365–382 (cit. on p. 5). 14

    work page 2010

  13. [13]

    Causal interpretation of stochastic differential equations

    Niels Hansen and Alexander Sokol. Causal interpretation of stochastic differential equations. In: Electronic Journal of Probability19 (Jan. 2014), pp. 1–24 (cit. on p. 6)

    work page 2014

  14. [14]

    Austin B. Hill. The Environment and Disease: Association or Causation? In:Proceedings of the Royal Society of Medicine(1965), pp. 295–300 (cit. on p. 2)

    work page 1965

  15. [15]

    Identification of Time-Dependent Causal Model: A Gaussian Process Treatment

    Biwei Huang, Kun Zhang, and Bernhard Schölkopf. Identification of Time-Dependent Causal Model: A Gaussian Process Treatment. In:Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence. 2015 (cit. on p. 2)

    work page 2015

  16. [16]

    Antti Hyttinen, Frederick Eberhardt, and Patrik O. Hoyer. Learning Linear Cyclic Causal Models with Latent Variables. In:Journal of Machine Learning Research13.109 (2012), pp. 3387–3439 (cit. on pp. 8, 20)

    work page 2012

  17. [17]

    Imbens and Donald B

    Guido W. Imbens and Donald B. Rubin. Causal Inference for Statistics, Social, and Biomedical Sciences. Cambridge University Press, 2015 (cit. on pp. 3, 6, 12, 20)

    work page 2015

  18. [18]

    Kitson, Anthony C

    Neville K. Kitson, Anthony C. Constantinou, Zhigao Guo, Yang Liu, and Kiattikun Chobtham. A survey of Bayesian Network structure learning. In:Artificial Intelligence Review(2023), pp. 1–94 (cit. on p. 12)

    work page 2023

  19. [19]

    Essai philosophique sur les probabilités

    Pierre Simon de Laplace. Essai philosophique sur les probabilités. Courcier, 1814, link to the 5th edition from 1825 (cit. on p. 6)

    work page

  20. [20]

    Cherng, and Joel T

    Hao-Chih Lee, Matteo Danieletto, Riccardo Miotto, Sarah T. Cherng, and Joel T. Dudley. Scaling structural learning with NO-BEARS to infer causal transcriptome networks. In:Pacific Symposium on Biocomputing 2020. World Scientific. 2019, pp. 391–402 (cit. on p. 2)

    work page 2020

  21. [21]

    Causal Structure Learning from Multivariate Time Series in Settings with Unmeasured Confounding

    Daniel Malinsky and Peter Spirtes. Causal Structure Learning from Multivariate Time Series in Settings with Unmeasured Confounding. In:Proceedings of 2018 ACM SIGKDD workshop on causal discovery. PMLR. 2018, pp. 23–47 (cit. on p. 6)

    work page 2018

  22. [22]

    Causal inference and causal explanation with background knowledge

    Christopher Meek. Causal inference and causal explanation with background knowledge. In:Proceed- ings of the Eleventh Conference on Uncertainty in Artificial Intelligence. UAI’95. Morgan Kaufmann Publishers Inc., 1995, pp. 403–410 (cit. on p. 21)

    work page 1995

  23. [23]

    Strong Completeness And Faithfulness In Bayesian Networks

    Christopher Meek. Strong Completeness And Faithfulness In Bayesian Networks. In:Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence. UAI’95. Morgan Kaufmann Publishers, 1995, pp. 411–418 (cit. on p. 21)

    work page 1995

  24. [24]

    Mooij, Dominik Janzing, Tom Heskes, and Bernhard Schölkopf

    Joris M. Mooij, Dominik Janzing, Tom Heskes, and Bernhard Schölkopf. On Causal Discovery with Cyclic Additive Noise Models. In:Advances in Neural Information Processing Systems (NeurIPS). Vol. 24. 2011 (cit. on p. 20)

    work page 2011

  25. [25]

    Mooij, Jonas Peters, Dominik Janzing, Jakob Zscheischler, and Bernhard Schölkopf

    Joris M. Mooij, Jonas Peters, Dominik Janzing, Jakob Zscheischler, and Bernhard Schölkopf. Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks. In:Journal of Machine Learning Research17.32 (2016), pp. 1–102 (cit. on pp. 2, 12)

    work page 2016

  26. [26]

    Dynamic bayesian networks: representation, inference and learning

    Kevin Patrick Murphy. Dynamic bayesian networks: representation, inference and learning. University of California, Berkeley, 2002 (cit. on p. 6)

    work page 2002

  27. [27]

    Nastl and Moritz Hardt

    Vivian Y. Nastl and Moritz Hardt. Do causal predictors generalize better to new domains? In: Advances in Neural Information Processing Systems. Vol. 37. 2024 (cit. on p. 13). 15

    work page 2024

  28. [28]

    Learning structures of Bayesian networks for variable groups

    Pekka Parviainen and Samuel Kaski. Learning structures of Bayesian networks for variable groups. In: International Journal of Approximate Reasoning88 (2017), pp. 110–127 (cit. on p. 21)

    work page 2017

  29. [29]

    Causal diagrams for empirical research

    Judea Pearl. Causal diagrams for empirical research. In:Biometrika 82.4 (1995), pp. 669–688 (cit. on pp. 3, 12)

    work page 1995

  30. [30]

    Causality: Models, Reasoning, and Inference

    Judea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, 2000 (cit. on p. 2)

    work page 2000

  31. [31]

    Causality: Models, Reasoning, and Inference

    Judea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, 2009 (cit. on pp. 2, 3, 5, 6, 9, 19, 21)

    work page 2009

  32. [32]

    The Book of Why: The New Science of Cause and Effect

    Judea Pearl and Dana Mackenzie. The Book of Why: The New Science of Cause and Effect. Basic Books, 2018 (cit. on p. 3)

    work page 2018

  33. [33]

    Causal Inference on Time Series using Restricted Structural Equation Models

    Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Causal Inference on Time Series using Restricted Structural Equation Models. In:Advances in Neural Information Processing Systems (NeurIPS). Vol. 26. 2013 (cit. on pp. 6, 7)

    work page 2013

  34. [34]

    Elements of Causal Inference: Foundations and Learning Algorithms

    Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of Causal Inference: Foundations and Learning Algorithms. MIT Press, 2017 (cit. on pp. 2, 5, 7, 19, 21)

    work page 2017

  35. [35]

    Joseph Ramsey, Madelyn Glymour, Ruben Sanchez-Romero, and Clark Glymour. A million variables and more: the Fast Greedy Equivalence Search algorithm for learning high-dimensional graphical causal models, with an application to functional magnetic resonance images. In:International Journal of Data Science and Analytics3 (2017), pp. 121–129 (cit. on p. 2)

    work page 2017

  36. [36]

    Reisach, Christof Seiler, and Sebastian Weichwald

    Alexander G. Reisach, Christof Seiler, and Sebastian Weichwald. Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy to Game. In:Advances in Neural Information Processing Systems (NeurIPS). Vol. 34. 2021 (cit. on p. 12)

    work page 2021

  37. [37]

    Reisach, Myriam Tami, Christof Seiler, Antoine Chambaz, and Sebastian Weichwald

    Alexander G. Reisach, Myriam Tami, Christof Seiler, Antoine Chambaz, and Sebastian Weichwald. A Scale-Invariant Sorting Criterion to Find a Causal Order in Additive Noise Models. In:Advances in Neural Information Processing Systems (NeurIPS). Vol. 36. 2023 (cit. on p. 12)

    work page 2023

  38. [38]

    Donald B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. In: Journal of Educational Psychology66.5 (1974), 688–701, link to the earlier open access 1972 ETS Technichal Report (cit. on pp. 2, 6)

    work page 1974

  39. [39]

    Causal network reconstruction from time series: From theoretical assumptions to practical estimation

    Jakob Runge. Causal network reconstruction from time series: From theoretical assumptions to practical estimation. In: Chaos: An Interdisciplinary Journal of Nonlinear Science28.7 (2018) (cit. on p. 6)

    work page 2018

  40. [40]

    Causality for Machine Learning

    Bernhard Schölkopf. Causality for Machine Learning. In:Probabilistic and Causal Inference: The Works of Judea Pearl. 2022, pp. 765–804 (cit. on pp. 10, 12)

    work page 2022

  41. [41]

    Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio

    Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan R. Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio. Toward Causal Representation Learning. In:Proceedings of the IEEE 109.5 (2021), pp. 612–634 (cit. on p. 12)

    work page 2021

  42. [42]

    Nonmonotonic reasoning and causation

    Yoav Shoham. Nonmonotonic reasoning and causation. In:Cognitive Science14.2 (1990), pp. 213–252 (cit. on p. 2)

    work page 1990

  43. [43]

    Complete Identification Methods for the Causal Hierarchy

    Ilya Shpitser and Judea Pearl. Complete Identification Methods for the Causal Hierarchy. In:Journal of Machine Learning Research9 (2008), pp. 1941–1979 (cit. on p. 3). 16

    work page 2008

  44. [44]

    Constructing Bayesian Network Models of Gene Expression Networks from Microarray Data

    Pater Spirtes, Clark Glymour, Richard Scheines, Stuart Kauffman, Valerio Aimale, and Frank Wimberly. Constructing Bayesian Network Models of Gene Expression Networks from Microarray Data. In: (2000) (cit. on p. 9)

    work page 2000

  45. [45]

    An Algorithm for Fast Recovery of Sparse Causal Graphs

    Peter Spirtes and Clark Glymour. An Algorithm for Fast Recovery of Sparse Causal Graphs. In: Social Science Computer Review9.1 (1991), pp. 62–72 (cit. on p. 13)

    work page 1991

  46. [46]

    Causation, Prediction, and Search

    Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search. In:Lecture Notes in Statistics81 (1993) (cit. on p. 2)

    work page 1993

  47. [47]

    Causation, Prediction, and Search

    Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search. MIT Press, 2001 (cit. on pp. 2, 8, 9, 21)

    work page 2001

  48. [48]

    Automated Search for Causal Relations: Theory and Practice

    Peter Spirtes, Clark Glymour, Richard Scheines, and Robert Tillman. Automated Search for Causal Relations: Theory and Practice. In:Heuristics, Probability and Causality: A Tribute to Judea Pearl (2010), pp. 467–502 (cit. on p. 5)

    work page 2010

  49. [49]

    On the Application of Probability Theory to Agricultural Experiments

    Jerzy Splawa-Neyman. On the Application of Probability Theory to Agricultural Experiments. In: Statistical Science5 (1932), pp. 465–480 (cit. on pp. 2, 6)

    work page 1932

  50. [50]

    Reply to Jim Woodward’s Comments on Wolfgang Spohn’s Laws of Belief

    Wolfgang Spohn. Reply to Jim Woodward’s Comments on Wolfgang Spohn’s Laws of Belief. In: Philosophy of Science86.4 (2019), pp. 773–784 (cit. on p. 5)

    work page 2019

  51. [51]

    The Laws of Belief: Ranking Theory and Its Philosophical Applications

    Wolfgang Spohn. The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press, 2012 (cit. on pp. 5, 9)

    work page 2012

  52. [52]

    The Structural Model and the Ranking Theoretic Approach to Causation: A Comparison

    Wolfgang Spohn. The Structural Model and the Ranking Theoretic Approach to Causation: A Comparison. In:Heuristics, Probability and Causality: A Tribute to Judea Pearl(2010), pp. 507–522 (cit. on p. 5)

    work page 2010

  53. [53]

    A Probabilistic Theory of Causality

    Patrick Suppes. A Probabilistic Theory of Causality. North-Holland Publishing Company, 1970 (cit. on p. 2)

    work page 1970

  54. [54]

    Foundations of causal discovery on groups of variables

    Jonas Wahl, Urmi Ninad, and Jakob Runge. Foundations of causal discovery on groups of variables. In: Journal of Causal Inference12.1 (2024), p. 20230041 (cit. on pp. 6, 21)

    work page 2024

  55. [55]

    Bayesian Nets and Causality: Philosophical and Computational Foundations

    Jon Williamson. Bayesian Nets and Causality: Philosophical and Computational Foundations. Oxford University Press, 2004 (cit. on p. 5)

    work page 2004

  56. [56]

    Making things happen: A theory of causal explanation

    James Woodward. Making things happen: A theory of causal explanation. Oxford university press, 2005 (cit. on p. 3)

    work page 2005

  57. [57]

    The Method of Path Coefficients

    Sewall Wright. The Method of Path Coefficients. In:The Annals of Mathematical Statistics5.3 (1934), pp. 161–215 (cit. on p. 2). 17 A Formal Definitions In this section we provide additional details regarding our formalism and definitions introduced in Section 3. For convenience, we will use the following notation: for anyk ∈ N∗, for any setS of cardinalit...

    work page 1934