信息幾何

信息幾何是用微分幾何研究概率論和統計學的交叉學科。^[1]

引言

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

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：

統計推斷^[8]
時間序列與線性系統
過濾問題^[9]
量子系統^[10]
神經網絡
機器學習
統計力學
生物學
統計學^[11] ^[12]
金融數學^[13]

另見

參考文獻

^ 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

[1] （頁面存檔備份，存於網際網路檔案館） Information Geometry journal by Springer
Information Geometry （頁面存檔備份，存於網際網路檔案館） overview by Cosma Rohilla Shalizi, July 2010
Information Geometry （頁面存檔備份，存於網際網路檔案館） notes by John Baez, November 2012
Information geometry for neural networks(pdf ) （頁面存檔備份，存於網際網路檔案館）, by Daniel Wagenaar

[1] 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）.

[2] 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.

[3] 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.

[4] 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）.

[5] Amari, Shun'ichi; Nagaoka, Hiroshi. Methods of Information Geometry. Translations of Mathematical Monographs 191. American Mathematical Society. 2000. ISBN 0-8218-0531-2.

[6] 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.

[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）.

[8] Kass, R. E.; Vos, P. W. Geometrical Foundations of Asymptotic Inference. Series in Probability and Statistics. Wiley. 1997. ISBN 0-471-82668-5.

[brigoieee-9] 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）.

[handel-10] 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.

[11] Amari, Shun'ichi. Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Berlin: Springer-Verlag. 1985. ISBN 0-387-96056-2.

[12] Murray, M.; Rice, J. Differential Geometry and Statistics. Monographs on Statistics and Applied Probability 48. Chapman and Hall. 1993. ISBN 0-412-39860-5.

[13] Marriott, Paul; Salmon, Mark (編). Applications of Differential Geometry to Econometrics. Cambridge University Press. 2000. ISBN 0-521-65116-6.

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