Applications of information geometry to the natural sciences

I am an information theorist and I try to give you my personal take. About three years ago, a conference on information geometry was held in Germany. As you can see there, there are lot of applications but most of the applications are deeply tied with Statistics. However, in my opinion, information geometry could not bring results that influence the related research in probability and statistics. And also no fundamentally original results was observed that excite researchers. Differential geometry is itself an obstruction due to the its complexity. Of course these are highly subjective judgments and no one can predict the future but at least currently, researchers are not so excited about the results.

You can have interesting connections with other discoveries of science but still, I think we have not seen some results from information geometry that are being widely used even in statistics.

I believe this is so, partly because the field is still very young and immature. I see a big potential in bring geometrical insights to statistics but I think that is very challenging for a young researcher.

I am looking forward to find a way to apply it in my own research though I have not found any way yet.

The section 2 and 3 of this book discusses the relation with statistics.

Personally, I worked in this area producing three papers appeared in conference proceedings (see this paper, this other and this one). I can provide you some book titles:

S. Amari&al., Differential Geometry in Statistical Inference, Institute of Mathematical Statistics (1987). This is currently free online here.

Shun-ichi Amari; Hiroshi Nagaoka, Methods of Information Geometry, AMS (2000).

Khadiga A. Arwini, Christopher T.J. Dodson, Information Geometry, Springer (2008).

You can also read the Wikipedia entry about this matter.

I think these hints should provide you a good starting point.

The question is an old one, but there is a new-ish book that would make a good answer: Information Theory and its Applications (2016) by Shun-Ichi Amari. (Book information on Springer's website | book on Amazon.com) It assumes pretty much the background you asked for, and half of the book covers applications. (Though it is more oriented to applications in machine learning than the natural sciences, generally speaking.)

Presumably you have long finished your undergrad thesis by now, but I hope this will be useful to future visitors with the same question.