Information Geometry
- 1st Edition, Volume 45 - September 26, 2021
- Imprint: North Holland
- Editors: Arni S.R. Srinivasa Rao, C.R. Rao, Angelo Plastino
- Language: English
- Hardback ISBN:9 7 8 - 0 - 3 2 3 - 8 5 5 6 7 - 9
- eBook ISBN:9 7 8 - 0 - 3 2 3 - 8 5 5 6 8 - 6
The subject of information geometry blends several areas of statistics, computer science, physics, and mathematics. The subject evolved from the groundbreaking article published by… Read more
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Request a sales quoteThe subject of information geometry blends several areas of statistics, computer science, physics, and mathematics. The subject evolved from the groundbreaking article published by legendary statistician C.R. Rao in 1945. His works led to the creation of Cramer-Rao bounds, Rao distance, and Rao-Blackawellization. Fisher-Rao metrics and Rao distances play a very important role in geodesics, econometric analysis to modern-day business analytics. The chapters of the book are written by experts in the field who have been promoting the field of information geometry and its applications.
- Written by experts for users of information geometry
- Basics to advanced readers are equally taken care
- Origins and Clarity on Foundations
Statisticians, computer scientists, physicists, econometricians and mathematicians
- Cover
- Title page
- Table of Contents
- Copyright
- Contributors
- Preface
- Section I: Foundations of information geometry
- Chapter 1: Revisiting the connection between Fisher information and entropy's rate of change
- Abstract
- 1: Introduction
- 2: Fisher information and Cramer–Rao inequality
- 3: Fisher information and the rate of change of Boltzmann–Gibbs entropy
- 4: Possible lines for future research
- 5: Conclusions
- References
- Chapter 2: Pythagoras theorem in information geometry and applications to generalized linear models
- Abstract
- 1: Introduction
- 2: Pythagoras theorems in information geometry
- 3: Power entropy and divergence
- 4: Linear regression model
- 5: Generalized linear model
- 6: Discussion
- References
- Further reading
- Chapter 3: Rao distances and conformal mapping
- Abstract
- 1: Introduction
- 2: Manifolds
- 3: Rao distance
- 4: Conformal mapping
- 5: Applications
- References
- Chapter 4: Cramer-Rao inequality for testing the suitability of divergent partition functions
- Abstract
- 1: Introduction
- 2: A first illustrative example
- 3: A Brownian motion example
- 4: The harmonic oscillator (HO) in Tsallis’ statistics
- 5: Failure of the Boltzmann-Gibbs (BG) statistics for Newton's gravitation
- 6: Statistics of gravitation in Tsallis’ statistics
- 7: Conclusions
- References
- Chapter 5: Information geometry and classical Cramér–Rao-type inequalities
- Abstract
- 1: Introduction
- 2: I-divergence and Iα-divergence
- 3: Information geometry from a divergence function
- 4: Information geometry for Bayesian CR inequality and Barankin bound
- 5: Information geometry for Bayesian α-CR inequality
- 6: Information geometry for Hybrid CR inequality
- 7: Summary
- Appendix
- References
- Section II: Theoretical applications and physics
- Chapter 6: Principle of minimum loss of Fisher information, arising from the Cramer-Rao inequality: Its role in evolution of bio-physical laws, complex systems and universes
- Abstract
- 1: Introduction (Fisher, 1922; Frieden 1998, 2004; Frieden and Gatenby, 2019)
- 2: Overview and comparisons of applications
- 3: Derivation of principle of maximum Fisher information (MFI)
- 4: Kantian view of Fisher information use to predict a physical law
- 5: Principle of minimum loss of Fisher information
- 6: Commonality of information-based growths of cancer and viral infections
- References
- Chapter 7: Quantum metrology and quantum correlations
- Abstract
- 1: Quantum correlations
- 2: Parameter estimation
- 3: Cramer–Rao bound
- 4: Quantum Fisher information
- 5: Quantum correlations in estimation theory
- 6: Conclusion
- References
- Chapter 8: Information, economics, and the Cramér-Rao bound
- Abstract
- 1: Introduction
- 2: Shannon entropy and Fisher information
- 3: Financial economics
- 4: Macroeconomics
- 5: Discussion and summary
- Acknowledgments
- References
- Chapter 9: Zipf's law results from the scaling invariance of the Cramer–Rao inequality
- Abstract
- 1: Introduction
- 2: Our goal
- 3: Fisher's information measure (FIM) and its minimization
- 4: Derivation of Zipf's law
- 5: Zipf plots
- 6: Summary
- References
- Further reading
- Section III: Advanced statistical theory
- Chapter 10: λ-Deformed probability families with subtractive and divisive normalizations
- Abstract
- 1: Introduction
- 2: λ-Deformation of exponential and mixture families
- 3: Deforming Legendre duality: λ-Duality
- 4: λ-Deformed entropy and divergence
- 5: Example: λ-Deformation of the probability simplex
- 6: Summary and conclusion
- References
- Chapter 11: Some remarks on Fisher information, the Cramer–Rao inequality, and their applications to physics
- Abstract
- 1: Introduction
- 2: Diffusion equation
- 3: Connection with Tsallis statistics
- 4: Conclusions
- Appendix
- References
- Index
- Edition: 1
- Volume: 45
- Published: September 26, 2021
- No. of pages (Hardback): 248
- No. of pages (eBook): 248
- Imprint: North Holland
- Language: English
- Hardback ISBN: 9780323855679
- eBook ISBN: 9780323855686
AS
Arni S.R. Srinivasa Rao
Arni S.R. Srinivasa Rao works in pure mathematics, applied mathematics, probability, and artificial intelligence and applications in medicine. He is a Professor at the Medical College of Georgia, Augusta University, U.S.A, and the Director of the Laboratory for Theory and Mathematical Modeling housed within the Division of Infectious Diseases, Medical College of Georgia, Augusta, U.S.A. Previously, Dr. Rao conducted research and/or taught at Mathematical Institute, University of Oxford (2003, 2005-07), Indian Statistical Institute (1998-2002, 2006-2012), Indian Institute of Science (2002-04), University of Guelph (2004-06). Until 2012, Dr. Rao held a permanent faculty position at the Indian Statistical Institute. He has won the Heiwa-Nakajima Award (Japan) and Fast Track Young Scientists Fellowship in Mathematical Sciences (DST, New Delhi). Dr. Rao also proved a major theorem in stationary population models, such as, Rao's Partition Theorem in Populations, Rao-Carey Theorem in stationary populations, and developed mathematical modeling-based policies for the spread of diseases like HIV, H5N1, COVID-19, etc. He developed a new set of network models for understanding avian pathogen biology on grid graphs (these were called chicken walk models), AI Models for COVID-19 and received wide coverage in the science media. Recently, he developed concepts such as “Exact Deep Learning Machines”, and “Multilevel Contours” within a bundle of Complex Number Planes.
Affiliations and expertise
Professor, Medical College of Georgia, USACR
C.R. Rao
book “Ancient Inhabitants of Jebel Moya” published by the Cambridge Press under the joint authorship of Rao and two anthropologists. On the basis of work done at CU during the two year period, 1946-1948, Rao earned a Ph.D. degree and a few years later Sc.D. degree of CU and the rare honor of life fellowship of Kings College, Cambridge.
He retired from ISI in 1980 at the mandatory age of 60 after working for 40 years during which period he developed ISI as an international center for statistical education and research. He also took an active part in establishing state statistical bureaus to collect local statistics and transmitting them to Central Statistical Organization in New Delhi. Rao played a pivitol role in launching undergraduate and postgraduate courses at ISI. He is the author of 475 research publications and several breakthrough papers contributing to statistical theory and methodology for applications to problems in all areas of human endeavor. There are a number of classical statistical terms named after him, the most popular of which are Cramer-Rao inequality, Rao-Blackwellization, Rao’s Orthogonal arrays used in quality control, Rao’s score test, Rao’s Quadratic Entropy used in ecological work, Rao’s metric and distance which are incorporated in most statistical books.
He is the author of 10 books, of which two important books are, Linear Statistical Inference which is translated into German, Russian, Czec, Polish and Japanese languages,and Statistics and Truth which is translated into, French, German, Japanese, Mainland Chinese, Taiwan Chinese, Turkish and Korean languages.
He directed the research work of 50 students for the Ph.D. degrees who in turn produced 500 Ph.D.’s. Rao received 38 hon. Doctorate degree from universities in 19 countries spanning 6 continents. He received the highest awards in statistics in USA,UK and India: National Medal of Science awarded by the president of USA, Indian National Medal of Science awarded by the Prime Minister of India and the Guy Medal in Gold awarded by the Royal Statistical Society, UK. Rao was a recipient of the first batch of Bhatnagar awards in 1959 for mathematical sciences and and numerous medals in India and abroad from Science Academies. He is a Fellow of Royal Society (FRS),UK, and member of National Academy of Sciences, USA, Lithuania and Europe. In his honor a research Institute named as CRRAO ADVANCED INSTITUTE OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE was established in the campus of Hyderabad University.
Affiliations and expertise
University of Hyderabad Campus, IndiaAP
Angelo Plastino
Angelo Plastino, PhD is Emeritus Professor of Physics, La Plata National University, Honorary Professor, University of Buenos Aires, Doctor Honoris Causa, University of Pretoria, Member of the Brazilian Academy of Sciences, Member of the Mexican Academy of Sciences, received the Scopus Elsevier Award to the most cited Argentine physicist in the period 1998-2008 and is the author of 680 refereed papers published in scientific journals (up to December 2020).
Affiliations and expertise
National University La Plata, La Plata, ArgentinaRead Information Geometry on ScienceDirect