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
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  and a harmonic trap .
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  on the simplex being a vector space).
The main point is that KL-divergence is not a metric, so imagining it as a distance in a space may give you some wrong intuitions. Its matrix of 2nd derivatives, the Fischer Information, works as a local metric, but then many people want to draw global pictures that try to extend this back to a global metric, which doesn't actually work.
"For inference, the only acceptable value for the Rényi-Tsallis parameter is α = 1, which is the correct information. That negates the generalisation to α != 1 which underlies Amari’s “α-divergences” in information geometry."
Have any IG proponents responded to or refuted Skilling's critiques? This is interesting because Shun'ichi Amari is credited amongst others with advancing the field in the 80's.
Can you imagine if GR metrics come from IG?
In the article, entanglement is a precise mathematical notion that is like an information based metric measuring the distance of a quantum state tensor (wavefunction) from the manifold of rank 1 tensors.
If you have more material, just put it for all.
While there seems like a potential well-balanced in-between to these complementing/contrasting philosophies and layers of view points, I feel partly well-suited to vent that from what I've seen, statistical data analysis seems to zealously want to gain understanding by using brute force, where self-ordering "shapes" simply want to flow which shows their Nature.