  Information geometry uses the formal tools of differential geometry to
describe the space of probability distributions as a Riemannian manifold with
an additional dual structure. The formal equivalence of compositional data with
discrete probability distributions makes it possible to apply the same
description to the sample space of Compositional Data Analysis (CoDA). The
latter has been formally described as a Euclidean space with an orthonormal
basis featuring components that are suitable combinations of the original
parts. In contrast to the Euclidean metric, the information-geometric
description singles out the Fisher information metric as the only one keeping
the manifold's geometric structure invariant under equivalent representations
of the underlying random variables. Well-known concepts that are valid in
Euclidean coordinates, e.g., the Pythogorean theorem, are generalized by
information geometry to corresponding notions that hold for more general
coordinates. In briefly reviewing Euclidean CoDA and, in more detail, the
information-geometric approach, we show how the latter justifies the use of
distance measures and divergences that so far have received little attention in
CoDA as they do not fit the Euclidean geometry favored by current thinking. We
also show how entropy and relative entropy can describe amalgamations in a
simple way, while Aitchison distance requires the use of geometric means to
obtain more succinct relationships. We proceed to prove the information
monotonicity property for Aitchison distance. We close with some thoughts about
new directions in CoDA where the rich structure that is provided by information
geometry could be exploited.
