信息几何

信息几何是用微分几何研究概率论和统计学的交叉学科。^[1]

从历史上看，信息几何可追溯到卡利安普迪·拉达克里希纳·拉奥的工作，他首先将费希尔矩阵视为黎曼度量。^[2]^[3]现代理论主要归功于甘利俊一，他的工作对该领域产生了重大影响。^[4]

经典的信息几何将有参概率模型视作黎曼流形。对于这类模型，可自然选择出黎曼度量，即费希尔信息度量。在概率模型为指数族时，有可能用黑塞度量（即凸函数的势给出的黎曼度量）导出统计流形，这时流形会自然继承两个平面仿射联络，以及正规布雷格曼散度。历史上，许多工作都致力于研究这些例子的相关几何。在现代背景下，信息几何适用于更广泛的背景，包括非指数族、非参数统计，甚至是不从已知概率模型导出的抽象统计流形。这些结果结合了信息论、仿射微分几何、凸分析等众多领域的技术。

该领域的标准参考书是甘利俊一与长冈浩司的《信息几何方法》^[5]及Nihat Ay等人的最新著作。^[6]Frank Nielsen在调查报告中做了较温和的介绍。^[7]2018年，《信息几何学》期刊正式创立，专门讨论该领域。

作为一个跨学科领域，信息几何已被广泛应用于各种领域，主要应用于统计分析、控制理论、神经网络、量子力学、信息论等领域。

下面是不完整的清单：

统计推断^[8]
时间序列与线性系统
过滤问题^[9]
量子系统^[10]
神经网络
机器学习
统计力学
生物学
统计学^[11] ^[12]
金融数学^[13]

鲁平几何
相对熵
随机几何
流形随机分析
投影滤波器

[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]. （原始内容存档 (PDF)于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. (编). 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. （原始内容存档于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]. （原始内容存档于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.
[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. （原始内容存档 (PDF)于2022-03-02）.
[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 (编). Applications of Differential Geometry to Econometrics. Cambridge University Press. 2000. ISBN 0-521-65116-6.

[1] （页面存档备份，存于互联网档案馆） Information Geometry journal by Springer
Information Geometry （页面存档备份，存于互联网档案馆） overview by Cosma Rohilla Shalizi, July 2010
Information Geometry （页面存档备份，存于互联网档案馆） notes by John Baez, November 2012
Information geometry for neural networks(pdf ) （页面存档备份，存于互联网档案馆）, by Daniel Wagenaar

[1] [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]. （原始内容存档 (PDF)于2023-11-09）.

[2] [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] [3]
Nielsen, F. Cramér-Rao Lower Bound and Information Geometry. Bhatia, R.; Rajan, C. S. (编). 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] [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. （原始内容存档于2023-11-09）.

[5] [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] [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] [7]
Nielsen, Frank. An Elementary Introduction to Information Geometry. Entropy. 2018, 22 (10) [2023-11-09]. （原始内容存档于2023-09-07）.

[8] [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] [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. （原始内容存档 (PDF)于2022-03-02）.

[handel-10] [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] [11]
Amari, Shun'ichi. Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. Berlin: Springer-Verlag. 1985. ISBN 0-387-96056-2.

[12] [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] [13]
Marriott, Paul; Salmon, Mark (编). Applications of Differential Geometry to Econometrics. Cambridge University Press. 2000. ISBN 0-521-65116-6.

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信息几何

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引言

应用

另见

参考文献

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