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At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information Geometry.

Having said that, I would like to be advised as to which books should I study in order to prepare myself to get acquainted to this subject.

More precisely, could someone tell me which books of differential geometry, probability and statistics should I read in order to introduce myself to it?

A progressive list of readings would be appreciated.

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    $\begingroup$ Why do you want to study information geometry? $\endgroup$
    – Christian Chapman
    Commented May 23, 2020 at 23:26
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    $\begingroup$ Actually, I am studying copulas at the moment, but I would like to explore its intersection with information geometry. Besides that (and more importantly) I am really interested in mathematical theories which have common applications within statistics. Thus I would like to receive book recommendations so that I can judge if I would keep studying it or not. $\endgroup$
    – user127351
    Commented May 23, 2020 at 23:41
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    $\begingroup$ This previous question might be helpful. mathoverflow.net/questions/344452/… $\endgroup$
    – Gabe K
    Commented May 24, 2020 at 0:07
  • $\begingroup$ @GabeK, thanks for the reference. But could you guide me through the literature by suggesting me some mathematical books which are the grounding for the subject? $\endgroup$
    – user127351
    Commented May 24, 2020 at 17:47
  • $\begingroup$ I'm not sure there is a textbook I would recommend that covers the prerequisites and also serves as an introduction to IG. For the geometry, I would recommend something like Do Carmo's textbook or John Lee's Riemannian manifolds (strictly speaking, these are two different topics but both are relevant for IG). I would also recommend learning some probability and statistics but I guess you are already familiar with those. $\endgroup$
    – Gabe K
    Commented May 25, 2020 at 12:31

1 Answer 1

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The lecture notes by Frank Nielsen are succinct and fairly self-contained and maybe good to get a first idea. The books [1,2] by Amari can serve for a more in-depth study and contain a fair bit of differential geometry background. In order to get a flavour for some of the applications, I would suggest [3].

EDIT: Some additional remarks on the differential geometry background

It is quite easy to get bogged down here. I would suggest first reading ref. 1 which has a nice overview of the most important concepts in chapter 1, and then reading additional texts as the need arises (see the comment by @Gabe below). In my view, it is important to have a good grasp of the fundamentals of smooth manifolds and have the geometric intuition for connections and metrics as additional structures on the manifold, i.e. what purpose they serve. It is good to have some experience in Riemannian geometry, enough to understand where and how dually flat connections are different from the Levi-Civita connection. Less important, at least for a first contact with information geometry, are all the global aspects, fibre bundles etc. Of course, if you are interested in those topics for their own sake, there is a lot of interesting stuff. Personally, I learned a lot from [4] but any book which covers the same range of topics would be fine, there are many texts which cater to different styles/preferences. For more guidance on geometry textbooks, there are a bunch of related questions here on MO.

  1. Amari, S., & Nagaoka, H. (2007). Methods of Information Geometry. American Mathematical Society.
  2. Amari, S. (2016). Information Geometry and Its Applications (Vol. 194). Tokyo: Springer Japan.
  3. Nielsen, Frank, Frank Critchley, and Christopher TJ Dodson. Computational Information Geometry. Springer: Berlin, Germany, 2017.
  4. Lee, J. M. (2009). Manifolds and Differential Geometry (Vol. 107). American Mathematical Society.

Please let me know if you need further or more specific suggestions.

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  • $\begingroup$ Thank you for the references @S.Surace. I am going to look for it. Could you also suggest me differential geometry books necessary to understand properly the subject? $\endgroup$
    – user127351
    Commented May 25, 2020 at 18:05
  • $\begingroup$ I added some remarks on the differential geometry background. $\endgroup$
    – S.Surace
    Commented May 26, 2020 at 8:39
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    $\begingroup$ This is a good answer but I would caution against using IG texts as a way of learning geometry. Information geometers have a very idiosyncratic approach to differential geometry and Lee is a much better reference for the basics. For instance, those lecture notes have a few inaccuracies with respect to differential geometry. These aren't important if you already know the subject, but would be quite misleading if you try to learn it from them. $\endgroup$
    – Gabe K
    Commented May 26, 2020 at 15:09
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    $\begingroup$ @Gabe, thanks for your comment, I fully agree. To be more precise, I suggest reading those IG texts to get a sense of what the relevant concepts are, and then going deeper with specialized texts. I'll refer to your comment in my answer. $\endgroup$
    – S.Surace
    Commented May 26, 2020 at 15:50

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