- 出版社: Springer; 1st ed. 2016 (2016年2月10日)
- 丛书名: Applied Mathematical Sciences
- 精装: 373页
- 语种： 英语
- ISBN: 4431559779
- 条形码: 0004431559779
- 商品尺寸: 15.6 x 2.2 x 23.4 cm
- 商品重量: 7.04 Kg
- ASIN: 4431559779
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Information Geometry and Its Applications (英语) 精装 – 2016年2月10日
“This book gives a reasonably accessible introduction to the subject and then considers various applications. … The book provides a nice introduction to the subject. … the book provides a nice introduction to a difficult subject that has many important applications.” (Marvin H. J. Gruber, Technometrics, Vol. 58 (4), April, 2016)
1 Manifold, Divergence and Dually Flat Structure.- 2 Exponential Families and Mixture Families of Probability.- 3 Invariant Geometry of Manifold of Probability.- 4 α-Geometry, Tsallis q-Entropy and Positive-Definite.- 5 Elements of Differential Geometry.- 6 Dual Affine Connections and Dually Flat Manifold.- 7 Asymptotic Theory of Statistical Inference.- 8 Estimation in the Presence of Hidden Variables.- 9 Neyman–Scott Problem.- 10 Linear Systems and Time Series.- 11 Machine Learning.- 12 Natural Gradient Learning and its Dynamics in Singular.- 13 Signal Processing and Optimization.- Index.
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From the previous literature "Methods of Information Geometry," extensive update has been made.
Far more references are added as sections / chapters, e.g. many recent results from after 2010 are cited and comprehensively explained.
This book should be the new standard as an introductory textbook for the field.
## Pythagorean theorem
There is a mistake in the statement (and proof) of theorem 1.2 (Generalized Pythagorean Theoreom).
The duality of the two geodesics are reversed in the statement (which you can confirm by proving the theorem).
That is, the dual geodesic PQ should be the geodesic (nabla), and the geodesic QR should be the dual geodesic (nabla-*)
A correct statement would be:
When triangle P Q R is orthogonal such that the geodesic connecting P and Q is orthogonal to the dual geodesic connecting Q and R,
D_\psi[P:R] = D_\psi[P:Q] + D_\psi[Q:R]
In the proof, (1.114) is the wrong part.
The dual version of the theorem (where PQ is dual geodesic and QR is the primal geodesic) is also wrong accordingly.
## Asymptotic properties of MLE
The part is not written precisely in the book (so the formulas don't make rigorous sense). I recommend reading Amari (1985) that is frequently cited in those sections if you want to follow the equations.