The relation between Granger causality and directed information theory: a review

TL;DR

This review explores the conceptual and theoretical connections between Granger causality and directed information theory, highlighting their relationships, differences, and applications in analyzing causal influences in stochastic processes.

Contribution

It provides a comprehensive synthesis of how directed information theory relates to and extends Granger causality, including decompositions and implications for multivariate analysis.

Findings

Directed information decomposes into transfer entropy and instantaneous coupling.

Mutual information can be expressed as sums of transfer entropy and instantaneous coupling.

Directed information theory naturally emerges in Granger causality hypothesis testing.

Abstract

This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The definitions of Granger causality based on prediction are recalled, and the importance of the observation set is discussed. We present the definitions based on conditional independence. The notion of instantaneous coupling is included in the definitions. The concept of Granger causality graphs is discussed. We present directed information theory from the perspective of studies of causal influences between stochastic processes. Causal conditioning appears to be the cornerstone for the relation between information theory and Granger causality. In the bivariate case, the fundamental measure is the directed information, which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling. We show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure, a relation known for the linear Gaussian case. We study the multivariate case, showing that the useful decomposition is blurred by instantaneous coupling. The links are further developed by studying how measures based on directed information theory naturally emerge from Granger causality inference frameworks as hypothesis testing.