Geometry and Statistics

1st Edition, Volume 46 - June 23, 2022
Editors: Frank Nielsen, Arni S.R. Srinivasa Rao, C.R. Rao
Language: English
Hardback ISBN:
9 7 8 - 0 - 3 2 3 - 9 1 3 4 5 - 4
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 1 3 4 6 - 1

Geometry and Statistics, Volume 46 in the Handbook of Statistics series, highlights new advances in the field, with this new volume presenting interesting chapters written by an… Read more

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Geometry and Statistics, Volume 46 in the Handbook of Statistics series, highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors.

Cover image
Title page
Table of Contents
Copyright
Contributors
Preface
Section I: Foundations in classical geometry and analysis
Chapter 1: Geometry, information, and complex bundles
Abstract
1: Introduction
2: Complex planes
3: Geometric analysis and Jordan curves
4: Summary
References
Chapter 2: Geometric methods for sampling, optimization, inference, and adaptive agents
Abstract
1: Introduction
2: Accelerated optimization
3: Hamiltonian-based accelerated sampling
4: Statistical inference with kernel-based discrepancies
5: Adaptive agents through active inference
Acknowledgments
References
Chapter 3: Equivalence relations and inference for sparse Markov models
Abstract
1: Introduction
2: Fitting SMMs and example applications
3: Equivalence relations and the computation of distributions of pattern statistics for SMMs
4: Summary
Acknowledgments
References
Section II: Information geometry
Chapter 4: Symplectic theory of heat and information geometry
Abstract
1: Preamble
2: Life and seminal work of Souriau on lie groups thermodynamics
3: From information geometry to lie groups thermodynamics
4: Symplectic structure of fisher metric and entropy as Casimir function in coadjoint representation
5: Covariant maximum entropy density by Souriau model
6: Conclusion
References
Further reading
Chapter 5: A unifying framework for some directed distances in statistics
Abstract
1: Divergences, statistical motivations, and connections to geometry
2: The framework
3: Aggregated/integrated divergences
4: Dependence expressing divergences
5: Bayesian contexts
6: Variational representations
7: Some further variants
Acknowledgments
References
Chapter 6: The analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions
Abstract
1: Introduction and motivation
2: Differential-geometric structures induced by smooth convex functions
3: Some illustrating examples
4: Information geometry of the mixture family of two distinct Cauchy distributions
5: Conclusion
Appendix: Symbolic computing notebook in MAXIMA
Acknowledgments
References
Chapter 7: Local measurements of nonlinear embeddings with information geometry
Abstract
1: Introduction
2: α-Divergence and autonormalizing
3: α-Discrepancy of an embedding
4: Empirical α-discrepancy
5: Connections to existing methods
6: Conclusion and extensions
Appendices
Appendix A: Proof of Lemma 1
Appendix B: Proof of Proposition 1
Appendix C: Proof of Proposition 2
Appendix D: Proof of Theorem 1
Acknowledgment
References
Section III: Advanced geometrical intuition
Chapter 8: Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac motion modeling
Abstract
1: Introduction
2: Parallel transport with ladder methods
3: Application to cardiac motion modeling
4: Conclusion
Acknowledgments
References
Chapter 9: Geometry and mixture models
Abstract
1: Introduction
2: Identification, singularities, and boundaries
3: Likelihood geometry
4: General geometric structures
5: Singular learning theory
6: Nonstandard testing problems
7: Discussion
References
Chapter 10: Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature
Abstract
1: Introduction
2: Gaussian distributions and RMT
3: Gaussian distributions and Bayesian inference
Appendix A: Riemannian symmetric spaces
Appendix B: Convex optimization
Appendix C: Proofs for Section B
References
Chapter 11: Multilevel contours on bundles of complex planes☆
Abstract
1: Introduction
2: Infinitely many bundles of complex planes
3: Multilevel contours in a random environment
4: Islands and holes in Bℝ(ℂ)
5: Concluding remarks
Acknowledgments
References
Index

Frank Nielsen

Frank Nielsen was awarded his PhD on adaptive computational geometry (1996) from INRIA/University of Cote d’Azur (France). He is a fellow of Sony Computer Science Laboratories Inc. (Sony CSL, Tokyo) where he currently conducts research on the fundamentals and practice of geometric machine learning and intelligence. He taught at Ecole Polytechnique (France) visual computing (Charles River Media, 2005) and high-performance computing for data science (Springer, 2016), and currently serves the following peer-reviewed journals: Springer Information Geometry, MDPI Entropy, and IEEE Transactions on Information Theory. Frank Nielsen co-organizes with Frederic Barbaresco the biannual conference Geometric Science of Information (GSI) since 2013.

Affiliations and expertise

Frank Nielsen is a fellow researcher of Sony Computer Science Laboratories Inc., Tokyo, Japan

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, USA

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, India

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Geometry and Statistics

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Frank Nielsen

Arni S.R. Srinivasa Rao

C.R. Rao