Causal conditioning and instantaneous coupling in causality graphs

TL;DR

This paper explores the relationship between Granger causality graphs and directed information theory, emphasizing the importance of both dynamical and instantaneous causality measures in multivariate systems, with theoretical insights and synthetic examples.

Contribution

It clarifies the roles of transfer entropy and instantaneous information exchange in multivariate causality graphs, highlighting their importance and limitations.

Findings

Decomposition of directed information into transfer entropy and information exchange does not hold in multivariate cases.

Both measures are essential for accurate causality inference in complex systems.

Synthetic examples demonstrate practical estimation challenges and theoretical distinctions.

Abstract

The paper investigates the link between Granger causality graphs recently formalized by Eichler and directed information theory developed by Massey and Kramer. We particularly insist on the implication of two notions of causality that may occur in physical systems. It is well accepted that dynamical causality is assessed by the conditional transfer entropy, a measure appearing naturally as a part of directed information. Surprisingly the notion of instantaneous causality is often overlooked, even if it was clearly understood in early works. In the bivariate case, instantaneous coupling is measured adequately by the instantaneous information exchange, a measure that supplements the transfer entropy in the decomposition of directed information. In this paper, the focus is put on the multivariate case and conditional graph modeling issues. In this framework, we show that the decomposition of directed information into the sum of transfer entropy and information exchange does not hold anymore. Nevertheless, the discussion allows to put forward the two measures as pillars for the inference of causality graphs. We illustrate this on two synthetic examples which allow us to discuss not only the theoretical concepts, but also the practical estimation issues.