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Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

Gijs van Seeventer, Saber Salehkaleybar

Abstract

We study identifiability in continuous-time linear stationary stochastic differential equations with known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrizations compatible with that structure. Under a notion of faithfulness, we derive criteria for characterising identifiability, non-identifiability, and partial identifiability for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

Abstract

We study identifiability in continuous-time linear stationary stochastic differential equations with known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrizations compatible with that structure. Under a notion of faithfulness, we derive criteria for characterising identifiability, non-identifiability, and partial identifiability for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.
Paper Structure (36 sections, 9 theorems, 125 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 36 sections, 9 theorems, 125 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Setup
  3. Background and Notation
  4. Stochastic Differential Equations
  5. Graphs
  6. Causal Interpretation of SDEs
  7. Definitions
  8. Sign Identification Problem
  9. Related Work
  10. Edge-Sign Identifiability Results
  11. Sign Identifiability in General Graphs
  12. Classical and Novel Graph Structures
  13. Without Latent Variables
  14. Latent Variables
  15. Numerical Results
  16. ...and 21 more sections

Key Result

Lemma 4.1

Let $e \in E$ be an edge in a graph $G=(V,E)$ and let $\Sigma \in F_G$. Then

Figures (1)

  • Figure 1: Nine considered causal structures. The red edge $\alpha$ under consideration is indicated by $\alpha$. Figures (a)-(g) are discussed in both Section \ref{['sec:classical_and_new_graph_structures']} and Section \ref{['sec:numerical_results']}. The node $H$ is considered observable and latent in Section \ref{['sec:no_latent']} and Section \ref{['sec:latent']}, respectively. Figures (h)-(j) are only studied numerically in Section \ref{['sec:numerical_results']}.

Theorems & Definitions (39)

  • Definition 2.1: M-Faithfulness
  • Remark 2.2
  • Definition 2.3: Edge Signature Set
  • Remark 2.4
  • Definition 2.5: Possible Set
  • Definition 2.6: Edge-Sign Identifiability
  • Remark 2.7
  • Definition 2.8: Pointwise Edge-Sign Identifiability
  • Remark 2.9
  • Lemma 4.1
  • ...and 29 more