Extremal Structural Causal Model (eSCM)

• eSCM is a specialized structural causal model that uses exponent measures to capture extreme event behaviors and tail dependencies.
• It employs activation variables and the single-big-jump principle to ensure that only one variable drives extreme responses, enhancing causal identifiability.
• The framework supports homogeneous structural functions and tail-based causal discovery, improving risk analysis in fields like finance and environmental studies.

An extremal structural causal model (eSCM) is a specialized formulation of structural causal models designed to capture the causal structure and dependencies that govern the behavior of random variables in the regime of extreme events. Unlike conventional SCMs, where the randomness is governed by full probability distributions, eSCMs employ exponent measures—homogeneous, infinite-mass laws rooted in multivariate extreme value theory—as their underlying randomness. This framework explicitly incorporates unique features of extremes, such as the single-big-jump principle and extremal conditional independence, and it supports model structures in which causal asymmetry can be leveraged for identification of causal directions in the tails of distributions.

1. Exponent Measures and the eSCM Structural Framework

The core mathematical object in an eSCM is the exponent measure Λ, which replaces the usual probabilistic description of noise. For a $d$-dimensional non-negative random vector $X$, if for some $\alpha > 0$,

$$t \mathbb{P}(t^{-1/\alpha} X \in B) \rightarrow \Lambda(B)\quad\text{as}\ t \to \infty$$

vaguely for Borel sets $B \subset [0,\infty)^d \setminus\{0\}$, then Λ is an exponent measure satisfying the scaling property

$\Lambda(c B) = c^{-\alpha} \Lambda(B)$

for any $c>0$. In the context of eSCMs, Λ embodies the asymptotic tail behavior of the structural equations' solutions and serves as the canonical law for the activation variables at each node.

The structural equations in an eSCM are formulated, for each node $v$ in a DAG $G=(V,E)$, as

$$Y_v = f_v(Y_{pa(v)}, \eta_v, \theta_v) = a_v \cdot \eta_v + h_v(Y_{pa(v)}, \theta_v)$$

where:

• $a_v > 0$ is an activation coefficient,
• $\eta_v$ is an activation variable distributed according to the (extremally independent) exponent measure Λ,
• $\theta_v$ is an auxiliary independent randomization variable (typically Uniform[0,1]),
• $h_v(\cdot, \cdot)$ is a homogeneous function in its first argument: $$h_v(c Y_{pa(v)}, \theta_v) = c h_v(Y_{pa(v)}, \theta_v)$$.

This model creates a separation between “activation” (representing the single big jump typical in multivariate extremes) and “proper structural propagation” (the deterministic/homogeneous influence of parents).

2. Activation Variables and the Single-Big-Jump Principle

The concept of the activation variable $\eta_v$ is foundational in eSCM construction. Activation variables are extremally independent: if an event $\eta_u > 0$ occurs for some $u$, then almost surely $\eta_{v} = 0$ for all $v \neq u$. This formalizes the single-big-jump principle, which postulates that, in the limit of large deviations, a single variable is responsible for the extreme event, with all others remaining negligible.

The structure imposed by the activation variables propagates through the entire model recursively: if $\eta_u$ alone is active, then for downstream nodes in the DAG, their values are computed solely through the deterministic effect of their active parent. This reflects how most mass of the exponent measure in the multivariate regime is concentrated on lower-dimensional faces of $[0,\infty)^d\setminus\{0\}$, i.e., the coordinate axes.

3. Proper Structural Functions and Homogeneity

A defining component of each node's structural assignment is the proper structural function $h_v(Y_{pa(v)}, \theta_v)$, which must satisfy scale-homogeneity. This property is essential for ensuring that the transformation of mass under scaling in the domain of the model aligns with the homogeneity property of the exponent measure. Deterministic propagation of extremes via $h_v$ mimics classical additive or multiplicative mechanisms in sums, max-linear, or other homogeneous models seen in extreme value theory.

Special cases:

• Sum-linear $h_v$ yields models akin to additive sum representations.
• Max-linear $h_v$ produces models where each node receives the maximum of parent effects, relevant for maximal risk propagation scenarios.

4. Causal Asymmetry and Direction Identifiability

A central problem in all causal modeling, and particularly in the extremal domain, is the identifiability of causal direction. eSCMs display a robust form of causal asymmetry under two natural assumptions:

• Nonzero activation ($a_v > 0$ for each $v$): each variable can itself be extreme.
• Nonzero parent effect: if the value of a parent is nonzero, its effect on its child is also nonzero via $h_v$.

These conditions induce a one-sided property in the exponent measure: for variables $u,v$, the mass $\Lambda_{\{u,v\}}(\{Y_u > 0, Y_v = 0\})$ is

• positive if $u$ is not an ancestor of $v$ in the DAG,
• zero if $u$ is an ancestor of $v$.

This property enables practical identifiability, as it is directly reflected in the support characteristics of the (bivariate) angular measure derived from the exponent measure. The angular asymmetry coefficient (AAC):

$\tau(u, v) = 1 - b - a$

(where $[a,b]$ is the minimal interval supporting the angular component when $Y$ is mapped to polar coordinates) quantifies this directionality: $\tau(u,v) > 0$ indicates $u \to v$, and $\tau(u,v) = -\tau(v,u)$ ensures skew symmetry. This quantification forms the basis for the EASE algorithm, which recovers causal order by estimating AACs from data.

5. Extremal Conditional Independence and Graphical Properties

eSCMs naturally encode extremal conditional independence, a generalization of classic conditional independence to the tail regime. In this framework, the exponent measure’s concentration on coordinate axes aligns with a generalization of d-separation, supporting directed graphical models in the extremal setting. When the structural equations are built as above, and the exponent measure of the joint vector $Y$ (the solution to the eSCM) admits a polar (angular) decomposition, the support and density of the angular component directly reflect the underlying directed graphical model.

All possible laws of directed graphical models under extremal conditional independence (as introduced in contemporary work) are encompassed by eSCMs of this formulation.

6. Algorithms and Applications

eSCMs facilitate causal structure learning in the tails via new algorithms that leverage causal asymmetry. The AAC-based global ordering procedure systematically estimates the causal ordering compatible with observed data in the extremes.

Simulation studies with sum-linear and max-linear eSCMs demonstrate that the AAC-based causal discovery outperforms or matches alternative approaches, such as causal tail coefficient (CTC) methods, in terms of ancestral violation rate. On real-world datasets (e.g., river discharge, benchmark cause-effect pairs), the eSCM framework recovers ground truth causal orderings when variables are appropriately heavy-tailed and transformed.

Key procedural steps:

• Data transformation to Pareto/regularly varying margins,
• Estimation of the exponent (angular) measure of the extremal sample,
• Computation and comparison of AACs for all variable pairs,
• Construction of the global causal order using EASE.

7. Relationship to Broader Extremal Causal Modeling and Limitations

eSCMs establish a rigorous foundation for causal analysis in the context of rare (extreme) events, extending the reach of SCMs to the tail regime, which is critical in risk assessment, resilience analysis, environmental extremes, and finance. They provide the mathematical and algorithmic tools for:

• Modeling rare event propagation in networks,
• Identifying causal relationships in the presence of single-big-jump dominance,
• Consistently recovering extremal causal DAGs from data under mild regularity assumptions.

Caveats include:

• The theoretical properties critically depend on correct specification of homogeneity and activation assumptions.
• Estimation of the exponent and angular measure in high dimensions remains a challenging open problem.
• Practical utility hinges on the system displaying clear extremal sparsity (single or few active big-jump variables).

Table: Distinction between Conventional SCMs and eSCMs

Aspect	Conventional SCM	Extremal SCM (eSCM)
Noise Law	Probability distribution	Exponent measure (infinite)
Structural Equation	$f_v(X_{pa(v)}, \epsilon_v)$	$f_v(Y_{pa(v)}, \eta_v, \theta_v)$
Independence	Standard conditional independence	Extremal conditional independence
Causality	All regimes	Tail regime (extremes)
Asymmetry/Identifiability	Generally ambiguous	Inherent via AAC, single-big-jump

The eSCM framework provides a comprehensive and mathematically rigorous extension of the SCM paradigm for the analysis and discovery of causal structure in the extremes, where the laws of dependence, propagation, and identifiability differ fundamentally from those in the bulk of the distribution, and where classical probabilistic and independence-based assumptions are replaced by the language and measure-theoretic machinery of multivariate regular variation and exponent measures (Fang et al., 1 Aug 2025).