Great citation. The title “An Elementary Introduction to Information Geometry” is a little silly. This is just a short introduction to information geometry, I don’t think there is anything more “advanced” floating around out there.

(I don’t know why the culture of pure math persists in this. I throw it in with proof by intimidation. Makes pure math types sound like they’re stuck intellectually/socially at being highly precocious 14 year olds. It’s not the individual, it’s the culture.)

Chentsov, that’s a name I haven’t heard in a long time. There’s a result that’s worth being generally known by the data analysis community. The article looks like it gets there more clearly than others I read a while ago.

Hear, hear! I think this problem is also beautifully put by the Knuth in Surreal Numbers. Results are boiled down to the smallest possible representation, without giving a historical/pedagogical overview, or even the slightest hint about how the author himself also struggled with the material.

This also boils down to what you understand the word "elementary" to mean. It is definitely not synonymous to "simple", rather as Feynman put it, "only requiring an infinite intelligence to understand".

I recall realizing as an undergraduate math major, the simpler the title of the book the more impossibly dense the material would be.

If your textbook was "Advanced X for Scientists and Engineers", easy-peasy. But if the textbook was "Introductory X" you were in for it.

I think it depends on the target audience. If it's math grad students that have had a first course of DG and probability and statistics, then it's elementary. I think the statement that it's self contained is practically incorrect, it's hard to believe that someone that doesn't already understand manifolds and connections can get a good enough intro to tackle IG from the overview in those notes.

Addendum: What I mean is that you can get much more abstract and take many more things for granted than it's done here. It's like reading an elementary introduction to the geometry of schemes, even if it's elementary there is a bunch of stuff that has to be assumed as known, even if the explanation of schemes per se is elementary

I think elementary is supposed to mean "self-contained" or "using the basic methods appropriate to the field", not necessarily "easy".