  Information theory provides a fundamental framework for the quantification of
information flows through channels, formally Markov kernels. However,
quantities such as mutual information and conditional mutual information do not
necessarily reflect the causal nature of such flows. We argue that this is
often the result of conditioning based on sigma algebras that are not
associated with the given channels. We propose a version of the (conditional)
mutual information based on families of sigma algebras that are coupled with
the underlying channel. This leads to filtrations which allow us to prove a
corresponding causal chain rule as a basic requirement within the presented
approach.
