Applications of information geometry to the natural sciences

Question

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (relatively) deep mathematical applications to biology or physics (I am not planning to pursue a Ph.D. in pure math).

To this extent, I recently stumbled across information geometry. By this I refer to the field of using data to generate a Riemannian manifold with the Fisher information metric. Could you tell me which applications the field has, particularly to (mathematical) biology or (statistical) physics?

Also, are there any good references, at the level of a Ph.D. student well-versed in probability, analysis, and geometry (but not as much so in statistical inference)?

(And, this goes a bit beyond the question, but if you have any other thoughts on what would be interesting subjects given my preferences above, please share!)

Answer 1

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.

Answer 2

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.