What's nice about this is that the derivation of a suitable metric allows us to compute trajectories that minimise quantities we care about (e.g. minimise energy dissipated), so this has clear potential to be useful. Some cool examples are in spin systems [2] and a harmonic trap [3].
For any differential geometers reading this: it seems to me that a good geometric way to think about this is as a fibre bundle, with the parameter space being the base space and the simplex being a vector bundle over it (see [4] on the simplex being a vector space).
[0] https://arxiv.org/pdf/0706.0559v2.pdf [1] https://arxiv.org/pdf/1201.4166.pdf [2] https://arxiv.org/pdf/1607.07425v1.pdf [3] http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041... [4] https://golem.ph.utexas.edu/category/2016/06/how_the_simplex...
As a results, it has a lot of practical application in machine learning and has been use successfully for classification in Neuroscience, Radar signal processing and computer vision.
we can also note that Information Geometry can be seen as a sub-field of Riemannian Geometry, with some equivalence between metric. For example, the cannonical metric for symetric and positive definite (SPD) matrices in Riemannian geometry is actually equivalent to the metric for multivariate normal distribution obtained with Information geometry.
For some application, IG is very efficient. it has been used for multivariate time-series for classification of EEG signal and was at the center of the winning solution of 3 kaggle challenges : https://github.com/alexandrebarachant