  Here we show an application of our recently proposed information-geometric
approach to compositional data analysis (CoDA). This application regards
relative count data, which are, e.g., obtained from sequencing experiments.
First we review in some detail a variety of necessary concepts ranging from
basic count distributions and their information-geometric description over the
link between Bayesian statistics and shrinkage to the use of power
transformations in CoDA. We then show that powering, i.e., the equivalent to
scalar multiplication on the simplex, can be understood as a shrinkage problem
on the tangent space of the simplex. In information-geometric terms,
traditional shrinkage corresponds to an optimization along a mixture (or m-)
geodesic, while powering (or, as we call it, exponential shrinkage) can be
optimized along an exponential (or e-) geodesic. While the m-geodesic
corresponds to the posterior mean of the multinomial counts using a conjugate
prior, the e-geodesic corresponds to an alternative parametrization of the
posterior where prior and data contributions are weighted by geometric rather
than arithmetic means. To optimize the exponential shrinkage parameter, we use
mean-squared error as a cost function on the tangent space. This is just the
expected squared Aitchison distance from the true parameter. We derive an
analytic solution for its minimum based on the delta method and test it via
simulations. We also discuss exponential shrinkage as an alternative to zero
imputation for dimension reduction and data normalization.
