Separation-based causal discovery for extremes

David Bolin; Jordan Richards; Junshu Jiang; Rapha\"el Huser

arxiv: 2505.08008 · v2 · submitted 2025-05-12 · 📊 stat.ME

Separation-based causal discovery for extremes

Junshu Jiang , Jordan Richards , Rapha\"el Huser , David Bolin This is my paper

Pith reviewed 2026-05-22 15:24 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal discoveryextreme valuesstructural causal modelstail dependenceDAG estimationconstraint-based algorithmspartial tail correlation

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The pith

A new class of structural causal models enables separation-based discovery of causal graphs for extreme values.

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

The paper defines XSCMs as a new class of structural causal models for extreme events that use transformed-linear algebra to represent causal relations. It proves that these models satisfy the Markov and faithfulness conditions with respect to partial tail uncorrelatedness, just as standard SCMs do for regular data. This means constraint-based causal discovery methods, which rely on separation in the graph corresponding to conditional independence, can now be used for tail behaviors. Sympathetic readers would care because extreme events in systems like hydrology or finance often have different dependence structures than average behavior, and this bridges the gap to apply proven algorithms there.

Core claim

We propose a new class of SCMs, called XSCMs, which leverage transformed-linear algebra to model causal relationships among extreme values. Similar to traditional SCMs, we prove that XSCMs satisfy the causal Markov and causal faithfulness properties with respect to partial tail (un)correlatedness. This enables estimation of the underlying DAG for extremes using separation-based tests, and makes many state-of-the-art constraint-based causal discovery algorithms directly applicable. We further consider undirected graph estimation for tail-dependent data and validate the approach on simulations up to 50 variables and real river discharge data.

What carries the argument

XSCMs defined via transformed-linear algebra, which carry the argument by preserving causal Markov and faithfulness properties for partial tail (un)correlatedness so that separation in the graph implies tail uncorrelation.

If this is right

Separation-based tests can estimate the DAG for extremes.
Many state-of-the-art constraint-based causal discovery algorithms become directly applicable to tail data.
Undirected graph estimation becomes feasible for relationships among tail-dependent and heavy-tailed variables.
The framework applies to large systems with up to 50 variables and to real data such as river discharges from the Danube basin.

Where Pith is reading between the lines

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

This could be combined with existing tail dependence estimators to improve graph recovery in very high dimensions.
Applying the same separation logic to other heavy-tailed domains such as climate extremes or financial crashes may uncover previously hidden causal structures.
A natural extension is to derive consistency rates for the estimated graphs when the number of variables grows with sample size.
Testing the method on data generated from known violations of the transformed-linear assumption would clarify its practical robustness.

Load-bearing premise

Causal relationships among extreme values can be represented via transformed-linear algebra while still preserving the Markov and faithfulness properties with respect to partial tail uncorrelatedness.

What would settle it

Generate synthetic data from a known DAG using an extreme value model outside the transformed-linear class and verify whether separation-based tests on partial tail correlations recover the correct edges or produce systematic errors.

read the original abstract

Structural causal models (SCMs), with an underlying directed acyclic graph (DAG), provide a powerful analytical framework to describe the interaction mechanisms in large-scale complex systems. However, when the system exhibits extreme events, the governing mechanisms can change dramatically, and SCMs with a focus on rare events are needed. We propose a new class of SCMs, called XSCMs, which leverage transformed-linear algebra to model causal relationships among extreme values. Similar to traditional SCMs, we prove that XSCMs satisfy the causal Markov and causal faithfulness properties with respect to partial tail (un)correlatedness. This enables estimation of the underlying DAG for extremes using separation-based tests, and makes many state-of-the-art constraint-based causal discovery algorithms directly applicable. We further consider the problem of undirected graph estimation for relationships among tail-dependent (and potentially heavy-tailed) data. The effectiveness of our method, compared to alternative approaches, is validated through simulation studies on large-scale systems with up to 50 variables, and in a well-studied application to river discharge data from the Danube basin. Finally, we apply the framework to investigate complex market-wide relationships in China's derivatives market.

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

XSCMs extend SCMs to extremes with transformed-linear equations and prove Markov plus faithfulness for partial tail uncorrelatedness, so standard separation algorithms apply directly.

read the letter

The main point is that the paper creates a new structural causal model tailored to extreme values. By using transformed-linear algebra, they define XSCMs and then prove that these models obey the causal Markov property and causal faithfulness when we look at partial tail uncorrelatedness instead of the usual conditional independence. This setup lets existing separation-based algorithms for causal discovery be applied straight to extreme data without major modifications. They handle the theory part effectively. The manuscript includes the necessary definitions, a recursive way to build the model, and the proofs for the two properties. There is no obvious gap or extra assumption that would invalidate the link to standard causal discovery methods. The simulations cover large systems with 50 variables and show comparisons to other approaches, while the real-data examples on the Danube basin and China's derivatives market add some applied relevance. One area that could use more attention is the modeling assumption itself. Transformed-linear relations provide a convenient structure, but extremes in fields like climate or finance can exhibit dependencies that are not well approximated this way, especially with heavy tails or sudden shifts. It would help to see more discussion on when this approximation holds and how the method performs under misspecification. The validation is there, but details on threshold selection for defining extremes and any uncertainty quantification in the tail measures would make the empirical results more convincing. Overall, this work sits at the intersection of causal inference and extreme value statistics. It should appeal to readers who need to uncover causal structures in tail-dependent data for risk assessment. The thinking is clear and the engagement with prior work on SCMs and extremes looks honest. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces XSCMs, a new class of structural causal models for extreme values based on transformed-linear algebra. It proves that these models satisfy the causal Markov and faithfulness properties with respect to partial tail (un)correlatedness, which justifies applying separation-based constraint algorithms for DAG estimation in extreme data. The framework is validated via simulations on systems with up to 50 variables and real-data examples including Danube river discharge and China's derivatives market.

Significance. If the proofs hold, the work provides a principled way to extend causal discovery to tail events, which is valuable for risk analysis in hydrology, finance, and other domains with heavy tails. The explicit establishment of Markov and faithfulness properties for the new dependence measure is a core strength, as is the demonstration that standard algorithms become directly usable. The large-scale simulation setup and two distinct real-world applications further support practical relevance.

major comments (2)

[§4] §4, Theorem on causal Markov property: the proof that partial tail uncorrelatedness implies d-separation in the XSCM graph relies on the transformed-linear representation commuting with the tail operator; an explicit verification for a three-node chain with heavy-tailed innovations would confirm that no additional non-degeneracy condition is required.
[Simulation section] Simulation section, 50-variable experiments: the reported superiority over baselines lacks per-replicate standard errors or bootstrap intervals on the structural Hamming distance, making it difficult to assess whether the performance gain is statistically reliable across the Monte Carlo runs.

minor comments (2)

[Abstract] The notation for partial tail uncorrelatedness is used throughout but first defined only in §2.2; adding a one-sentence reminder in the abstract or introduction would improve accessibility.
[Application section] In the Danube application, the choice of threshold for defining extremes and the resulting sample size after thresholding should be stated explicitly to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and positive evaluation of our work on XSCMs for causal discovery in extreme values. We address each of the major comments in detail below and outline the revisions we will make to the manuscript.

read point-by-point responses

Referee: [§4] §4, Theorem on causal Markov property: the proof that partial tail uncorrelatedness implies d-separation in the XSCM graph relies on the transformed-linear representation commuting with the tail operator; an explicit verification for a three-node chain with heavy-tailed innovations would confirm that no additional non-degeneracy condition is required.

Authors: We appreciate the referee's suggestion to strengthen the proof of the causal Markov property. While the general proof in Theorem 4.1 relies on the properties of the transformed-linear algebra and its commutation with the tail operator, we agree that an explicit check for a simple three-node chain would be helpful to illustrate that no additional non-degeneracy conditions are needed even with heavy-tailed innovations. In the revised manuscript, we will include such a verification example in Section 4. revision: yes
Referee: [Simulation section] Simulation section, 50-variable experiments: the reported superiority over baselines lacks per-replicate standard errors or bootstrap intervals on the structural Hamming distance, making it difficult to assess whether the performance gain is statistically reliable across the Monte Carlo runs.

Authors: We agree that reporting variability across replicates would allow readers to better assess the statistical reliability of the performance improvements. In the revised version of the manuscript, we will add per-replicate standard errors (or bootstrap confidence intervals) for the structural Hamming distance in the 50-variable simulation experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit definitions and proofs

full rationale

The paper introduces XSCMs by defining a transformed-linear structural equation model for extremes, then supplies a recursive construction and direct proofs that these models satisfy the causal Markov property and faithfulness with respect to partial tail (un)correlatedness. These steps are independent of any fitted parameters or self-referential assumptions; the properties follow from the model definition in the same manner as standard SCMs, without reducing the central claim to an input by construction. The manuscript is self-contained against external benchmarks for the stated theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the new definition of XSCMs and the two transferred causal properties; these introduce domain assumptions about tail dependence that are not supplied by upstream literature.

axioms (2)

domain assumption Causal Markov property holds for XSCMs with respect to partial tail (un)correlatedness
Invoked to justify use of separation-based tests for DAG recovery.
domain assumption Causal faithfulness property holds for XSCMs with respect to partial tail (un)correlatedness
Invoked to ensure that separation in the graph corresponds to tail independence.

invented entities (1)

XSCMs no independent evidence
purpose: Structural causal models specialized for extreme values via transformed-linear algebra
New model class defined in the paper to capture causal mechanisms among extremes.

pith-pipeline@v0.9.0 · 5733 in / 1459 out tokens · 43363 ms · 2026-05-22T15:24:18.201889+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

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

We propose a new class of SCMs, called XSCMs, which leverage transformed-linear algebra to model causal relationships among extreme values. ... prove that XSCMs satisfy the causal Markov and causal faithfulness properties with respect to partial tail (un)correlatedness.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Definition 3.1 (Extremal structural causal model (XSCM)) ... Xi = (αi◦Zi)⊕ ⨁_{Xj∈Pa(Xi)} (βj→i◦Xj)

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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.