  This paper addresses the problem of inferring circulation of information
between multiple stochastic processes. We discuss two possible frameworks in
which the problem can be studied: directed information theory and Granger
causality. The main goal of the paper is to study the connection between these
two frameworks. In the case of directed information theory, we stress the
importance of Kramer's causal conditioning. This type of conditioning is
necessary not only in the definition of the directed information but also for
handling causal side information. We also show how directed information
decomposes into the sum of two measures, the first one related to Schreiber's
transfer entropy quantifies the dynamical aspects of causality, whereas the
second one, termed instantaneous information exchange, quantifies the
instantaneous aspect of causality. After having recalled the definition of
Granger causality, we establish its connection with directed information
theory. The connection is particularly studied in the Gaussian case, showing
that Geweke's measures of Granger causality correspond to the transfer entropy
and the instantaneous information exchange. This allows to propose an
information theoretic formulation of Granger causality.
