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Structural Causal Models for Extremes: an Approach Based on Exponent Measures

Shuyang Bai, Fei Fang, Tiandong Wang

Abstract

We introduce a new formulation of structural causal models for extremes, called the extremal structural causal model (eSCM). Unlike conventional structural causal models, where randomness is governed by a probability distribution, eSCMs use an exponent measure, an infinite-mass law that naturally arises in the analysis of multivariate extremes. Central to this framework are activation variables, which abstract the single-big-jump principle, along with additional randomization that enriches the class of eSCM laws. This formulation encompasses all possible laws of directed graphical models under the recently introduced notion of extremal conditional independence. We also identify an inherent asymmetry in eSCMs under natural assumptions, enabling the identifiability of causal directions, a central challenge in causal inference. Finally, we propose a method that utilizes this causal asymmetry and demonstrate its effectiveness in both simulated and real datasets.

Structural Causal Models for Extremes: an Approach Based on Exponent Measures

Abstract

We introduce a new formulation of structural causal models for extremes, called the extremal structural causal model (eSCM). Unlike conventional structural causal models, where randomness is governed by a probability distribution, eSCMs use an exponent measure, an infinite-mass law that naturally arises in the analysis of multivariate extremes. Central to this framework are activation variables, which abstract the single-big-jump principle, along with additional randomization that enriches the class of eSCM laws. This formulation encompasses all possible laws of directed graphical models under the recently introduced notion of extremal conditional independence. We also identify an inherent asymmetry in eSCMs under natural assumptions, enabling the identifiability of causal directions, a central challenge in causal inference. Finally, we propose a method that utilizes this causal asymmetry and demonstrate its effectiveness in both simulated and real datasets.

Paper Structure

This paper contains 30 sections, 14 theorems, 131 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Extremal structural causal models
  3. Background on multivariate extremes and exponent measure
  4. The formulation of extremal structural casual model
  5. Basic properties of the Law of eSCM
  6. Examples
  7. Approximation of eSCMs by probablistic SCMs
  8. Interventions of eSCM
  9. Extremal causal Markov condition
  10. Extremal causal asymmetry and causal direction learning
  11. Extremal causal asymmetry
  12. Statistical identification of extremal causal direction
  13. Extremal causal order identification
  14. Numerical results
  15. Simulation studies of extremal causal order discovery
  16. ...and 15 more sections

Key Result

Proposition 1

Following the construction in Definition def:eSCM, we have where $s>0$ is as in eq:Lambda_perp, $\mathsf{E}_{\boldsymbol{\theta}}$ denotes the expectation with respect to $\mathsf{P}_{\boldsymbol{\theta}}$. In addition, the law $\Lambda=\mathcal{L}(\mathbf{Y})$ is an exponent measure that satisfies eq:Lambda homo and eq:Lambda marginal with $s_v$ as in eq:s

Figures (7)

  • Figure 1: Illustration of DAG notation. $\mathrm{pa}(2)=\{1,5\}$. $\mathrm{an}(3)=\{1,2,5\}$. $\mathrm{An}(3)=\{1,2,3,5\}$. $\mathrm{de}(3)=\{4,6\}$. $\mathrm{De}(3)=\{3,4,6\}$. $\mathrm{nd}(3)=\{1,2,5\}$. The sub-DAG $\mathcal{A}(4)$ consists of the node set $\mathrm{An}(4)=\{1,2,3,4,5\}$ and the edge set $\{(1\rightarrow 2), (5\rightarrow 2), (2\rightarrow 3), (3\rightarrow 4)\}$. The sub-DAG $\mathcal{A}_2(4)$ consists of the node set $\mathrm{An}_2(4)=\{2,3,4\}$ and the edge set $\{(2\rightarrow 3), (3\rightarrow 4)\}$. $\mathrm{An}_2^{\circ}(4)=\{3,4\}$.
  • Figure 2: Illustration of the law of $(Y_1,Y_2)$ in \ref{['eq:Y_1 -> Y_2']} when $\beta$ is fixed (left) v.s. when it randomized (right). A thick solid line denotes a mass concentration, whereas the shaded cone illustrates randomization.
  • Figure 3: Illustration of intervening an eSCM with the DAG in Figure \ref{['fig:DAG illu']}. The new node $7$ represents the intervention node $d+1$ in Definition \ref{['Def:interv']}. The original edge $(2\to 3)$ is erased. When node $3$ is intervened to be a nonzero value through conditioning on a positive value of the activation variable $\eta_7$, the nodes $1$, $2$ and $5$ in the dashed box become unobservable and take values $0$.
  • Figure 4: Illustration of angular support interval $[a,b]$. The shaded area represents the smallest cone/sector containing the support of $\Lambda_{\{u,v\}}$.
  • Figure 5: Behavior of angular asymmetry coefficient (AAC) with respect to causal relations under Assumptions \ref{['ass:eta act']} and \ref{['ass:asym']}. Solid lines indicate measure masses, while shaded cones represent angular supports.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3: eSCM
  • Remark 1
  • Proposition 1
  • Example 1
  • Example 2
  • Theorem 1
  • Definition 4
  • Definition 5
  • ...and 19 more