信息幾何
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/340px-Normal_Distribution_PDF.svg.png)
引言
編輯從歷史上看,信息幾何可追溯到卡利安普迪·拉達克里希納·拉奧的工作,他首先將費希爾矩陣視為黎曼度量。[2][3]現代理論主要歸功於甘利俊一,他的工作對該領域產生了重大影響。[4]
經典的信息幾何將有參概率模型視作黎曼流形。對於這類模型,可自然選擇出黎曼度量,即費希爾信息度量。在概率模型為指數族時,有可能用黑塞度量(即凸函數的勢給出的黎曼度量)導出統計流形,這時流形會自然繼承兩個平面仿射聯絡,以及正規布雷格曼散度。歷史上,許多工作都致力於研究這些例子的相關幾何。在現代背景下,信息幾何適用於更廣泛的背景,包括非指數族、非參數統計,甚至是不從已知概率模型導出的抽象統計流形。這些結果結合了信息論、仿射微分幾何、凸分析等眾多領域的技術。
該領域的標準參考書是甘利俊一與長岡浩司的《信息幾何方法》[5]及Nihat Ay等人的最新著作。[6]Frank Nielsen在調查報告中做了較溫和的介紹。[7]2018年,《信息幾何學》期刊正式創立,專門討論該領域。
應用
編輯作為一個跨學科領域,信息幾何已被廣泛應用於各種領域,主要應用於統計分析、控制理論、神經網絡、量子力學、信息論等領域。
下面是不完整的清單:
另見
編輯參考文獻
編輯- ^ Nielsen, Frank. The Many Faces of Information Geometry (PDF). Notices of the AMS (American Mathematical Society). 2022, 69 (1): 36-45 [2023-11-09]. (原始內容存檔 (PDF)於2023-11-09).
- ^ Rao, C. R. Information and Accuracy Attainable in the Estimation of Statistical Parameters. Bulletin of the Calcutta Mathematical Society. 1945, 37: 81–91. Reprinted in Breakthroughs in Statistics. Springer. 1992: 235–247. S2CID 117034671. doi:10.1007/978-1-4612-0919-5_16.
- ^ Nielsen, F. Cramér-Rao Lower Bound and Information Geometry. Bhatia, R.; Rajan, C. S. (編). Connected at Infinity II: On the Work of Indian Mathematicians. Texts and Readings in Mathematics. Special Volume of Texts and Readings in Mathematics (TRIM). Hindustan Book Agency. 2013: 18–37. ISBN 978-93-80250-51-9. S2CID 16759683. arXiv:1301.3578 . doi:10.1007/978-93-86279-56-9_2.
- ^ Amari, Shun'ichi. A foundation of information geometry. Electronics and Communications in Japan. 1983, 66 (6): 1–10 [2023-11-09]. doi:10.1002/ecja.4400660602. (原始內容存檔於2023-11-09).
- ^ Amari, Shun'ichi; Nagaoka, Hiroshi. Methods of Information Geometry. Translations of Mathematical Monographs 191. American Mathematical Society. 2000. ISBN 0-8218-0531-2.
- ^ Ay, Nihat; Jost, Jürgen; Lê, Hông Vân; Schwachhöfer, Lorenz. Information Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 64. Springer. 2017. ISBN 978-3-319-56477-7.
- ^ Nielsen, Frank. An Elementary Introduction to Information Geometry. Entropy. 2018, 22 (10) [2023-11-09]. (原始內容存檔於2023-09-07).
- ^ Kass, R. E.; Vos, P. W. Geometrical Foundations of Asymptotic Inference. Series in Probability and Statistics. Wiley. 1997. ISBN 0-471-82668-5.
- ^ Brigo, Damiano; Hanzon, Bernard; LeGland, Francois. A differential geometric approach to nonlinear filtering: the projection filter (PDF). IEEE Transactions on Automatic Control. 1998, 43 (2): 247–252 [2023-11-09]. doi:10.1109/9.661075. (原始內容存檔 (PDF)於2022-03-02).
- ^ van Handel, Ramon; Mabuchi, Hideo. Quantum projection filter for a highly nonlinear model in cavity QED. Journal of Optics B: Quantum and Semiclassical Optics. 2005, 7 (10): S226–S236. Bibcode:2005JOptB...7S.226V. S2CID 15292186. arXiv:quant-ph/0503222 . doi:10.1088/1464-4266/7/10/005.
- ^ Amari, Shun'ichi. Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Berlin: Springer-Verlag. 1985. ISBN 0-387-96056-2.
- ^ Murray, M.; Rice, J. Differential Geometry and Statistics. Monographs on Statistics and Applied Probability 48. Chapman and Hall. 1993. ISBN 0-412-39860-5.
- ^ Marriott, Paul; Salmon, Mark (編). Applications of Differential Geometry to Econometrics. Cambridge University Press. 2000. ISBN 0-521-65116-6.
外部連結
編輯- [1] (頁面存檔備份,存於互聯網檔案館) Information Geometry journal by Springer
- Information Geometry (頁面存檔備份,存於互聯網檔案館) overview by Cosma Rohilla Shalizi, July 2010
- Information Geometry (頁面存檔備份,存於互聯網檔案館) notes by John Baez, November 2012
- Information geometry for neural networks(pdf ) (頁面存檔備份,存於互聯網檔案館), by Daniel Wagenaar