A Sample Path Measure of Causal Influence

TL;DR

This paper introduces a sample path dependent measure of causal influence between time series, enabling identification of specific data patterns that indicate information flow, with applications to stationary and time-varying processes.

Contribution

It proposes a novel causal measure based on sample paths, connects it with directed information, and develops estimation methods with finite sample guarantees.

Findings

The measure can identify causality patterns in data.

Sequential prediction improves estimation accuracy.

The approach extends to time-varying causal scenarios.

Abstract

We present a sample path dependent measure of causal influence between two time series. The proposed measure is a random variable whose expected sum is the directed information. A realization of the proposed measure may be used to identify the specific patterns in the data that yield a greater flow of information from one process to another, even in stationary processes. We demonstrate how sequential prediction theory may be leveraged to obtain accurate estimates of the causal measure at each point in time and introduce a notion of regret for assessing the performance of estimators of the measure. We prove a finite sample bound on this regret that is determined by the regret of the sequential predictors used in obtaining the estimate. We estimate the causal measure for a simulated collection of binary Markov processes using a Bayesian updating approach. Finally, given that the measure is a function of time, we demonstrate how estimators of the causal measure may be extended to effectively capture causality in time-varying scenarios.