Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

2024-02-28

arXiv.org

AI Summary

This paper addresses causal structure inference for stochastic dynamical systems—such as those in science, health, and finance—modeled by stochastic differential equations (SDEs). We formulate time-oriented interval-process conditional independence (CI) constraints applicable to both fully and partially observed settings, and establish, for the first time, an equivalence between the directed acyclic dependence graph induced by an SDE and these CI constraints over interval-coordinate processes. We further propose the first statistically consistent CI test for continuous-path data based on the signature kernel, integrating path signature theory, stochastic process embedding, and causal discovery algorithms. This approach overcomes fundamental limitations of existing CI tests in continuous-time settings and ensures unique identifiability of the ancestral graph. Theoretical guarantees are rigorously provided, and empirical evaluation across multiple benchmark SDEs and generalized stochastic process models demonstrates significant improvements over state-of-the-art methods.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Security and Privacy: Data transparency and provenance

Abstract

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via"which variables enter the differential of which other variables". In this paper, we develop conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model. We then provide a sound and complete causal discovery algorithm, capable of handling both fully and partially observed data, and uniquely recovering the underlying or induced ancestral graph by exploiting time directionality assuming a CI oracle. Finally, to make our algorithm practically usable, we also propose a flexible, consistent signature kernel-based CI test to infer these constraints from data. We extensively benchmark the CI test in isolation and as part of our causal discovery algorithms, outperforming existing approaches in SDE models and beyond.

Problem

• Develops CI constraints for stochastic processes modeled by SDEs.
• Provides a causal discovery algorithm for fully and partially observed data.
• Proposes a signature kernel-based CI test for practical causal inference.

Innovation

• Develops CI constraints for SDE-induced graphs
• Proposes sound, complete causal discovery algorithm
• Introduces signature kernel-based CI test