  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.
