TITLE: Information geometry: geometry of exponential families
QUESTION [7 upvotes]: I've read in various locations that the geometry of exponential families are flat.
Is this true? I don't understand because, I have also read that the family of gaussians with unknown mean and unknown variance has hyperbolic geometry, which is not flat! and the family of multinomial distributions has spherical geometry.
Here is a arxiv paper which seems to suggest both are true:
https://arxiv.org/abs/0911.4863
There must be something fundamental that I'm not understanding. I'm not an expert in differential geometry.

REPLY [4 votes]: After doing a little more research, I've figured it out.
exponential families are not necessary flat under the Levi-Civita connection generated by the fisher information metric.
But they are flat under other connections, namely the e-connection and the m-connection. which are defined in the arxiv paper I linked to above.