Conditional Local Independence Testing for It\^o processes with Applications to Dynamic Causal Discovery

TL;DR

This paper introduces a novel hypothesis test for conditional local independence in Itô processes, enabling causal inference in dynamical systems, with applications demonstrated in brain fMRI data.

Contribution

We develop a new statistical test based on semimartingale decomposition and martingale properties for causal discovery in Itô processes, with proven consistency and practical validation.

Findings

Test accurately detects local independence in simulated data.

Method successfully applied to real brain fMRI data.

Proven consistency and reliable level and power of the test.

Abstract

Inferring causal relationships from dynamical systems is the central interest of many scientific inquiries. Conditional local independence, which describes whether the evolution of one process is influenced by another process given additional processes, is important for causal learning in such systems. In this paper, we propose a hypothesis test for conditional local independence in It\^o processes. Our test is grounded in the semimartingale decomposition of the It\^o process, with which we introduce a stochastic integral process that is a martingale under the null hypothesis. We then apply a test for the martingale property, quantifying potential deviation from local independence. The test statistics is estimated using the optimal filtering equation. We show the consistency of the estimation, thereby establishing the level and power of our test. Numerical verification and a real-world application to causal discovery in brain resting-state fMRIs are conducted.