  The decomposition of channel information into synergies of different order is
an open, active problem in the theory of complex systems. Most approaches to
the problem are based on information theory, and propose decompositions of
mutual information between inputs and outputs in se\-veral ways, none of which
is generally accepted yet.
  We propose a new point of view on the topic. We model a multi-input channel
as a Markov kernel. We can project the channel onto a series of exponential
families which form a hierarchical structure. This is carried out with tools
from information geometry, in a way analogous to the projections of probability
distributions introduced by Amari. A Pythagorean relation leads naturally to a
decomposition of the mutual information between inputs and outputs into terms
which represent single node information, pairwise interactions, and in general
n-node interactions.
  The synergy measures introduced in this paper can be easily evaluated by an
iterative scaling algorithm, which is a standard procedure in information
geometry.
