信息几何:修订间差异
删除的内容 添加的内容
加入{{Expand language}}標記到條目 |
内容扩充 |
||
第1行: | 第1行: | ||
[[File:Normal Distribution PDF.svg|340px|thumb|right|所有正态分布的集合构成具有[[双曲几何]]的统计流形。]] |
|||
{{Expand language|1=en|time=2020-01-19T13:58:29+00:00}} |
|||
⚫ | |||
'''信息几何'''是用[[微分几何]]研究[[概率论]]和[[统计学]]的交叉学科。<ref>{{cite journal | first=Frank | last= Nielsen | |
|||
{{Geometry-stub}} |
|||
title=The Many Faces of Information Geometry |
|||
| journal = Notices of the AMS |
|||
| publisher= American Mathematical Society |
|||
| year= 2022| |
|||
volume = 69 | number= 1| |
|||
page=36-45| url=https://www.ams.org/journals/notices/202201/rnoti-p36.pdf }}</ref> |
|||
这门学科研究[[统计流形]],即点与[[概率分布]]相对应的[[黎曼流形]]。 |
|||
== 引言 == |
|||
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[[费希尔信息|费希尔矩阵]]视为[[黎曼流形|黎曼度量]]。<ref>{{cite journal |last=Rao |first=C. R. |year=1945 |title=Information and Accuracy Attainable in the Estimation of Statistical Parameters |journal=Bulletin of the Calcutta Mathematical Society |volume=37 |pages=81–91 }} Reprinted in {{cite book |title=Breakthroughs in Statistics |publisher=Springer |year=1992 |pages=235–247 |doi=10.1007/978-1-4612-0919-5_16 |s2cid=117034671 }}</ref><ref>{{cite book |first=F. |last=Nielsen |year=2013 |arxiv=1301.3578 |chapter=Cramér-Rao Lower Bound and Information Geometry |title=Connected at Infinity II: On the Work of Indian Mathematicians |series=Texts and Readings in Mathematics |editor-first=R. |editor1-last=Bhatia |editor2-first=C. S. |editor2-last=Rajan |volume=Special Volume of Texts and Readings in Mathematics (TRIM) |pages=18–37 |publisher=Hindustan Book Agency |doi=10.1007/978-93-86279-56-9_2 |isbn=978-93-80250-51-9 |s2cid=16759683 }}</ref>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。<ref>{{cite journal |first=Shun'ichi |last=Amari | title=A foundation of information geometry | journal=Electronics and Communications in Japan |year=1983 |volume=66 |issue=6 |pages=1–10 |doi=10.1002/ecja.4400660602 | url=https://onlinelibrary.wiley.com/doi/abs/10.1002/ecja.4400660602}}</ref> |
|||
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博的《信息几何方法》<ref>{{cite book |first1=Shun'ichi |last1=Amari |first2=Hiroshi |last2=Nagaoka |title=Methods of Information Geometry |series=Translations of Mathematical Monographs |volume=191 |publisher=American Mathematical Society |year=2000 |isbn=0-8218-0531-2 }}</ref>及Nihat Ay等人的最新著作。<ref>{{cite book |first1=Nihat |last1=Ay |first2=Jürgen |last2=Jost |author-link2=Jürgen Jost |first3=Hông Vân |last3=Lê |first4=Lorenz |last4=Schwachhöfer |title=Information Geometry |volume=64 |series=Ergebnisse der Mathematik und ihrer Grenzgebiete |publisher=Springer |year=2017 |isbn=978-3-319-56477-7 }}</ref>Frank Nielsen在调查报告中做了较温和的介绍。<ref>{{cite journal|first=Frank |last=Nielsen |journal = Entropy | title=An Elementary Introduction to Information Geometry |date=2018 |url =https://www.mdpi.com/1099-4300/22/10/1100 | volume =22 | number =10 }}</ref>2018年,《信息几何学》期刊正式创立,专门讨论该领域。 |
|||
== 应用 == |
|||
⚫ | |||
下面是不完整的清单: |
|||
* 统计推断<ref>{{cite book |first1=R. E. |last1=Kass |first2=P. W. |last2=Vos |year=1997 |title=Geometrical Foundations of Asymptotic Inference |series=Series in Probability and Statistics |publisher=Wiley |isbn=0-471-82668-5 }}</ref> |
|||
* 时间序列与线性系统 |
|||
* 过滤问题<ref name="brigoieee">{{cite journal |last1=Brigo |first1=Damiano |last2=Hanzon |first2=Bernard | last3= LeGland | first3 = Francois | year=1998 |title=A differential geometric approach to nonlinear filtering: the projection filter |journal= IEEE Transactions on Automatic Control |volume= 43 |issue=2 |pages= 247–252 |doi=10.1109/9.661075 |url=https://hal.inria.fr/hal-02101519/file/rr-2598.pdf |author1-link=Damiano_Brigo }}</ref> |
|||
* 量子系统<ref name="handel">{{cite journal |last1=van Handel |first1=Ramon |last2=Mabuchi |first2=Hideo |year=2005 |title=Quantum projection filter for a highly nonlinear model in cavity QED | journal= Journal of Optics B: Quantum and Semiclassical Optics| volume= 7 |issue=10 |pages=S226–S236 |doi=10.1088/1464-4266/7/10/005 |arxiv=quant-ph/0503222 |bibcode=2005JOptB...7S.226V |s2cid=15292186 }}</ref> |
|||
* 神经网络 |
|||
* 机器学习 |
|||
* 统计力学 |
|||
* 生物学 |
|||
* 统计学<ref>{{cite book |first=Shun'ichi |last=Amari |year=1985 |title=Differential-Geometrical Methods in Statistics |series=Lecture Notes in Statistics |publisher=Springer-Verlag |location=Berlin |isbn=0-387-96056-2 }}</ref> <ref>{{cite book |first1=M. |last1=Murray |first2=J. |last2=Rice |year=1993 |title=Differential Geometry and Statistics |series=Monographs on Statistics and Applied Probability |volume=48 |publisher=[[Chapman and Hall]] |isbn=0-412-39860-5 }}</ref> |
|||
* 金融数学<ref>{{cite book |editor-first=Paul |editor-last=Marriott |editor2-first=Mark |editor2-last=Salmon |title=Applications of Differential Geometry to Econometrics |publisher=Cambridge University Press |year=2000 |isbn=0-521-65116-6 }}</ref> |
|||
==另见== |
|||
* [[鲁平几何]] |
|||
* [[相对熵]] |
|||
* [[随机几何]] |
|||
* [[流形随机分析]] |
|||
* [[投影滤波器]] |
|||
==参考文献== |
|||
{{Reflist}} |
|||
== 外部链接 == |
|||
* [https://www.springer.com/mathematics/geometry/journal/41884] Information Geometry journal by Springer |
|||
* [http://bactra.org/notebooks/info-geo.html Information Geometry] overview by Cosma Rohilla Shalizi, July 2010 |
|||
* [http://math.ucr.edu/home/baez/information/ Information Geometry] notes by [[John C. Baez|John Baez]], November 2012 |
|||
* [http://www.its.caltech.edu/~daw/papers/98-Wage2.pdf Information geometry for neural networks(pdf )], by Daniel Wagenaar |
|||
{{Differentiable computing}} |
|||
[[Category:微分几何]] |
[[Category:微分几何]] |
2023年11月9日 (四) 03:42的版本
引言
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博的《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。
应用
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:
另见
参考文献
- ^ Nielsen, Frank. The Many Faces of Information Geometry (PDF). Notices of the AMS (American Mathematical Society). 2022, 69 (1): 36-45.
- ^ 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. (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.
- ^ Amari, Shun'ichi. A foundation of information geometry. Electronics and Communications in Japan. 1983, 66 (6): 1–10. doi:10.1002/ecja.4400660602.
- ^ 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).
- ^ 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. doi:10.1109/9.661075.
- ^ 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 (编). Applications of Differential Geometry to Econometrics. Cambridge University Press. 2000. ISBN 0-521-65116-6.
外部链接
- [1] Information Geometry journal by Springer
- Information Geometry overview by Cosma Rohilla Shalizi, July 2010
- Information Geometry notes by John Baez, November 2012
- Information geometry for neural networks(pdf ), by Daniel Wagenaar
|