Let $\mathcal{M} = \{p_\theta := p(\cdot | \theta), \theta \in \Theta\}$ be a statistical manifold with Fisher information metric: $$g_{{jk}}(\theta )=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta _{i}}}\log p(X;\theta )\right)\left({\frac {\partial }{\partial \theta _{j}}}\log p(X;\theta )\right) \right].$$

The Wikipedia article on the topic derives the metric form Euclidean metric by changing variables. I can understand the procedure but I have questions related to the flatness of $\mathcal{M}$. In Amari's book; "Information Geometry and its application", it is said that such manifold is flat (dually flat actually) so

1- Is the above derivation enough to conclude that the manifold is flat (I mean the fact that the metric is derived from the Euclidean metric)?

2- Is there a straightforward way to show that the curvature is 0 everywhere?


This is a common source of confusion.

The flatness in information geometry refers to the dual (with respect to the Fisher-Rao metric) connections $\nabla$ and $\nabla^*$, called exponential and mixture connections, and not to the metric or Levi-Civita connection. For the latter, please see Bruveris, M., & Michor, P. W. (2019). Geometry of the Fisher-Rao metric on the space of smooth densities on a compact manifold. Mathematische Nachrichten, 292(3), 511–523. https://doi.org/10.1002/mana.201600523

Finite-dimensional statistical manifolds can have negative curvature, for example the information manifold of 1-dimensional Gaussians is isometric to the Poincaré half-plane.


You just have to read Jean-Louis Koszul Work who has developed elementary structure of Information Geometry. You will find main Koszul references in the paper "Jean-Louis Koszul and the elementary structures of Information Geometry": https://www.academia.edu/attachments/56029291/download_file?st=MTUyNTYzMDQ1NiwxOTIuNTQuMTQ0LjIyOSw0OTY2MjA3Mw%3D%3D&s=swp-toolbar

Fisher metric could be extended on homogeneous manifolds or on Lie group by Fisher-Souriau metric developed in the framework of Souriau Lie Group Thermodynamics: [A] Marle, C.-M. From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics. Entropy 2016, 18, 370. http://www.mdpi.com/1099-4300/18/10/370/pdf [B] Barbaresco, F. Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families. Entropy 2016, 18, 386. http://www.mdpi.com/1099-4300/18/11/386/pdf

Other references: GSI’17 Geometric Science of Information Third International Conference, GSI 2017, Paris, France, November 7-9, 2017, Proceedings Editors: Nielsen, Frank, Barbaresco, Frédéric (Eds.) http://www.springer.com/us/book/9783319684444 Videos: https://www.youtube.com/channel/UCnE9-LbfFRqtaes49cN2DVg/videos

CIRM seminar TGSI'17 on Topological & Geometrical Structures of Information http://forum.cs-dc.org/category/94/tgsi2017

Differential Geometrical Theory of Statistics Frédéric Barbaresco and Frank Nielsen (Eds.) Pages: XIV, 458, Published: 6 June 2017 (This book is a printed edition of the Special Issue Differential Geometrical Theory of Statistics that was published in Entropy) Free Download: http://www.mdpi.com/books/pdfdownload/book/313/1

Information, Entropy and Their Geometric Structures Frédéric Barbaresco and Ali Mohammad-Djafari (Eds.) Pages: XXIV, 528, Published: 1 September 2015 (This book is a printed edition of the Special Issue Information, Entropy and Their Geometric Structures that was published in Entropy) Free Download: http://www.mdpi.com/books/pdfdownload/book/127/1

  • $\begingroup$ Thanks for the great refs. I guess somebody down voted (wasn't me) because it wasn't exactly answer to my questions. Nevertheless, they are valuable for someone like me who has just started to do research on this topic. $\endgroup$
    – RozaTh
    May 6 '18 at 20:55

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