Causal conditioning and instantaneous coupling in causality graphs

In this paper, we develop the link between Granger causality graphs and directed information theory. In the bivariate case we show that directed information splits into two terms, transfer entropy and instantaneous information exchange, that may be used to assess dynamical causality and instantaneous coupling. We extend the analysis to the multivariate case, for which the notion of causal conditioning encompasses two different situations. This is due to the existence of two possible definitions for instantaneous coupling, one leading to independence graphs, the other leading to the more well accepted conditional independence graphs. We provide the decomposition of the directed information in terms of measures that may be used to infer causality graphs. Estimation and testing procedures are detailed, and used to illustrate our point on a four dimensional example.