An Asymmetric Independence Model for Causal Discovery on Path Spaces

2025-03-12

AI Summary

This work addresses causal discovery for nonlinear, dynamic systems governed by stochastic differential equations (SDEs) with feedback loops and partial observability—scenarios where standard acyclic causal models fail. Method: We establish a rigorous correspondence between E-separation in directed mixed graphs (DMGs) and conditional independence among coordinate processes on SDE path space, introducing the first asymmetric independence model closed under both cycles and marginalization. Contribution/Results: We prove the global Markov property for SDE path spaces; characterize the maximal element of the equivalence class of DMGs—whose skeleton and v-structures are uniquely determined by E-separation; show identifiability of the maximal graph under full observation; and verify existence of such maximal elements under partial observation in four-node systems. This framework provides the first graph-based causal model for nonlinear, dynamic, latent-variable systems that is both theoretically sound and computationally tractable.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We develop the theory linking 'E-separation' in directed mixed graphs (DMGs) with conditional independence relations among coordinate processes in stochastic differential equations (SDEs), where causal relationships are determined by"which variables enter the governing equation of which other variables". We prove a global Markov property for cyclic SDEs, which naturally extends to partially observed cyclic SDEs, because our asymmetric independence model is closed under marginalization. We then characterize the class of graphs that encode the same set of independence relations, yielding a result analogous to the seminal 'same skeleton and v-structures' result for directed acyclic graphs (DAGs). In the fully observed case, we show that each such equivalence class of graphs has a greatest element as a parsimonious representation and develop algorithms to identify this greatest element from data. We conjecture that a greatest element also exists under partial observations, which we verify computationally for graphs with up to four nodes.

Problem

• Link E-separation in DMGs with SDEs' conditional independence relations.
• Prove global Markov property for cyclic and partially observed SDEs.
• Identify greatest graph element in equivalence classes from data.

Innovation

• Links E-separation in DMGs with SDEs
• Proves global Markov property for cyclic SDEs
• Develops algorithms to identify greatest graph element