Information geometry:
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| Format: | Buch |
| Sprache: | English |
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Amsterdam
North-Holland, an imprint of Elsevier
[2021]
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| Schriftenreihe: | Handbook of statistics
volume 45 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 231 Seiten Illustrationen, Diagramme 24 cm |
| ISBN: | 9780323855679 |
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| adam_text | Contents Contributors Preface x¡ xiii Section 1 Foundations of information geometry 1. Revisiting the connection between Fisher information and entropy s rate of change 1 3 A.R. Plastino, Angelo Plastino, and F. Pennini 1. 2. 3. Introduction Fisher information and Cramer-Rao inequality Fisher information and the rate of change ofBoltzmann-Gibbs entropy 3.1 3.2 2. Brownian particle with constant drag force Systems described by an N-dimensional Fokker-Planck equation 3 4 6 10 11 4. Possible lines for future research 5. Conclusions References 12 13 13 Pythagoras theorem in information geometry and applications to generalized linear models 15 Shinto Eguchi 3. 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 16 18 24 31 35 40 41 42 Rao distances and conformal mapping 43 Arni S.R. Srinivasa Rao and Steven C. Krantz 1. 2. Introduction Manifolds 43 44 2.1 46 Conformality between two regions V
VI 4. Contents 3. Rao distance 4. Conforma! mapping 5. Applications Acknowledgments References 49 53 55 55 Cramer-Rao inequality for testing the suitability of divergent partition functions 57 Angelo Plastino, Marlo Carlos Rocca, and Diana Monteoliva 1. Introduction 2. A first illustrative example 2.1 Evaluation of the partition function 2.2 Instruction manual for using our procedure 2.3 Evaluation of (r) 2.4 Dealing with (r2) 2.5 Obtaining fisher information measure 2.6 The six steps to obtain a finite Fisher s information 2.7 Cramer-Rao inequality (CRI) 2.8 Numerical example 3. A Brownian motion example 3.1 The present partition function 3.2 Mean values of x-powers 3.3 Tackling fisher 3.4 The present Cramer-Rao inequality 4. The harmonic oscillator (HO) in Tsallis statistics 4.1 The HO-Tsallis partition function 4.2 HO-Tsallis mean values for r2 57 58 59 61 61 62 62 63 64 64 65 65 66 66 67 68 68 69 5. 6. 4.3 Mean value of r 4.4 Variance V 4.5 The HO-Tsallis Fisher information measure Failure of the Boltzmann-Gibbs (BG) statistics for Newton s gravitation 5.1 Tackling Zv 5.2 Mean values derived from our partitionfunction (PP) 5.3 Variance Δγ= {і2)—{ή2 5.4 Gravitational FIM 5.5 Incompatibility between Boltzmann-Gibbs statistics (BGS) and long-range interactions Statistics of gravitation in Tsallis statistics 6.1 Gravity-Tsallis partition function 6.2 Gravity-Tsallis mean values for rand r2 Tsallis Gravity treatment and Fisher s information measure 6.4 Tsallis Gravity treatment and Cramer-Rao inequality (CRI) 7. Conclusions References 46 69 69 70 70 70 71 72 73 73 73 73
74 6.3 76 77 77 78
Contents 5. vj¡ Information geometry and classical Cramér-Rao-type inequalities 79 Kumar Vijay Mishra and M. Ashok Kumar 1. 2. 3. Introduction /-divergence and /„-divergence 79 82 2.1 2.2 2.3 84 84 85 Extension to infinite X Bregman vs Csiszár Classical vs quantum CR inequality Information geometry from a divergence function 86 3.1 3.2 3.3 88 90 95 Information geometry for a-CR inequality An а-version of Cramér֊Rao inequality Generalized version of Cramér-Rao inequality 4. Information geometry for Bayesian CR inequality and Barankin bound 98 5. Information geometry for Bayesian a-CR inequality 101 6. Information geometry for Hybrid CR inequality 106 7. Summary 106 Acknowledgments 107 Appendix 107 A.1 Other generalizations of Cramér-Rao inequality References Section II Theoretical applications and physics 6. 107 110 115 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 117 B. Roy Frieden 1. Introduction 1.1 On learning, energy, sensory messages 1.2 On variational approaches 1.3 Vital role played by information 118 119 119 2. Overview and comparisons of applications 2.1 2.2 2.3 2.4 2.5 2.6 Classical dynamics Quantum physics Biology Thermodynamics Extending use of the principle ofnatural selection From biological cell to earth to solar system, galaxy, universe, and multiverse 2.7 Creation of a multiverse by requiring its Fisher I to be maximized 2.8 Analogy of a cancer universe 2.9 What ultimately causes a multiverse to form? 2.10 Is there empirical evidence for
amultiverse having formed? 118 120 120 120 121 121 122 123 124 125 125 126
viii Contents 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 Details of the process of growing successive universes How many universes N might exist in the multiverse? Annihilation of universes Growth of a bubble of nothing Counter-growth of new universes Possibility of many annihilation waves How large a number N of universes exist? Is the multiverse merely a theoretical construct? Should the fact that we do not, and have not observed life elsewhere in our universe affect a belief that we exist in a multiverse? 3. Derivation of principle of maximum Fisher information (MFI) 3.1 Cramer-Rao (C-R) inequality 3.2 On derivation of the C-R inequality 3.3 What do such data values (augmented by knowledge of a single equality obeyed by the system physics) have to say about the unknown system physics? 4. Kantian view of Fisher information use to predict a physical law 4.1 How principle of maximum information originates with Kant 4.2 On significance of the information difference l—J 5. Principle of minimum loss of Fisher information 5.1 Verifying that minimum loss is actually achieved by the principle 5.2 Summary and foundations of the Fisher approach to knowledge acquisition 5.3 What is accomplished by use of the Fisher approach 6. Commonality of information-based growths of cancer and viral infections 6.1 MFI applied to early cancer growth 6.2 Later-stage cancer growth 6.3 MFI applied to early covid-19 growth 6.4 Common biological causes of cancer- and covid-19 growth; the ACE2 link References 7. Quantum metrology and quantum correlations 126 128 129 129 130 130 130 131 131 132 132
132 133 135 135 135 136 137 137 140 142 142 143 143 144 146 149 Diego G. Bussandri and Pedro W. Lamberti 1. 2. 3. 4. 5. Quantum correlations Parameter estimation Cramer-Rao bound Quantum Fisher information Quantum correlations in estimation theory 5.1 Heisenberg limit 5.2 Interferometric power 6. Conclusion References 149 152 153 155 156 157 159 160 160
Contents 8. IX Information, economics, and the Cramér-Rao bound 161 Raymond J. Hawkins and B. Roy Frieden 1. 2. 3. Introduction Shannon entropy and Fisher information Financial economics 161 162 164 3.1 3.2 164 168 Discount factors and bonds Derivative securities 4. Macroeconomics 5. Discussion and summary Acknowledgments References 9. Zipfs law results from the scaling invariance of the Cramer-Rao inequality 171 174 175 175 179 Alberto Hernando and Angelo Plastino 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 179 180 180 180 181 183 183 183 Section 111 Advanced statistical theory 10. Я-Deformed probability families with subtractive and divisive normalizations 185 187 /un Zhang and Ting-Kam Leonard Wong 1. 2. 3. Introduction 187 1.1 1.2 1.3 189 191 192 Deformation models Deformed probability families: General approach Chapter outline A-Deformation of exponential and mixture families 193 2.1 2.2 2.3 2.4 2.5 193 194 195 196 197 A-Deformation Deformation: Subtractive approach Deformation: Divisive approach Relation between the two normalizations А-Exponential and A-mixture families Deforming Legendre duality: A-Duality 199 3.1 3.2 3.3 199 201 3.4 From Bregman divergence to A-logarithmic divergence А-Deformed Legendre duality Relationship between А-conjugation and Legendre conjugation Information geometry of A-logarithmic divergence 202 206
x Contents 4. A-Deformedentropy and divergence 4.1 Relation between potential functions and Rényi entropy 4.2 Relation between A-Iogarithmic divergence and Rényi divergence 4.3 Entropy maximizing property of A-exponential family 5. Example: A-Deformation of the probability simplex 5.1 А-Exponential representation 5.2 A-Mixture representation 6. Summary and conclusion References 11. 207 208 209 210 210 211 212 214 Some remarks on Fisher information, the Cramer-Rao inequality, andtheir applicationsto physics 217 H.C. Miller, Angelo Plastino, and A.R. Plastino I. Introduction 2. Diffusion equation 3. Connection with Tsallis statistics 4. Conclusions Appendix A.1 The Cramer-Rao bound References Index 207 217 220 222 225 226 226 227 229
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| adam_txt |
Contents Contributors Preface x¡ xiii Section 1 Foundations of information geometry 1. Revisiting the connection between Fisher information and entropy's rate of change 1 3 A.R. Plastino, Angelo Plastino, and F. Pennini 1. 2. 3. Introduction Fisher information and Cramer-Rao inequality Fisher information and the rate of change ofBoltzmann-Gibbs entropy 3.1 3.2 2. Brownian particle with constant drag force Systems described by an N-dimensional Fokker-Planck equation 3 4 6 10 11 4. Possible lines for future research 5. Conclusions References 12 13 13 Pythagoras theorem in information geometry and applications to generalized linear models 15 Shinto Eguchi 3. 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 16 18 24 31 35 40 41 42 Rao distances and conformal mapping 43 Arni S.R. Srinivasa Rao and Steven C. Krantz 1. 2. Introduction Manifolds 43 44 2.1 46 Conformality between two regions V
VI 4. Contents 3. Rao distance 4. Conforma! mapping 5. Applications Acknowledgments References 49 53 55 55 Cramer-Rao inequality for testing the suitability of divergent partition functions 57 Angelo Plastino, Marlo Carlos Rocca, and Diana Monteoliva 1. Introduction 2. A first illustrative example 2.1 Evaluation of the partition function 2.2 Instruction manual for using our procedure 2.3 Evaluation of (r) 2.4 Dealing with (r2) 2.5 Obtaining fisher information measure 2.6 The six steps to obtain a finite Fisher's information 2.7 Cramer-Rao inequality (CRI) 2.8 Numerical example 3. A Brownian motion example 3.1 The present partition function 3.2 Mean values of x-powers 3.3 Tackling fisher 3.4 The present Cramer-Rao inequality 4. The harmonic oscillator (HO) in Tsallis' statistics 4.1 The HO-Tsallis partition function 4.2 HO-Tsallis mean values for r2 57 58 59 61 61 62 62 63 64 64 65 65 66 66 67 68 68 69 5. 6. 4.3 Mean value of r 4.4 Variance V 4.5 The HO-Tsallis Fisher information measure Failure of the Boltzmann-Gibbs (BG) statistics for Newton's gravitation 5.1 Tackling Zv 5.2 Mean values derived from our partitionfunction (PP) 5.3 Variance Δγ= {і2)—{ή2 5.4 Gravitational FIM 5.5 Incompatibility between Boltzmann-Gibbs statistics (BGS) and long-range interactions Statistics of gravitation in Tsallis' statistics 6.1 Gravity-Tsallis partition function 6.2 Gravity-Tsallis mean values for rand r2 Tsallis' Gravity treatment and Fisher's information measure 6.4 Tsallis' Gravity treatment and Cramer-Rao inequality (CRI) 7. Conclusions References 46 69 69 70 70 70 71 72 73 73 73 73
74 6.3 76 77 77 78
Contents 5. vj¡ Information geometry and classical Cramér-Rao-type inequalities 79 Kumar Vijay Mishra and M. Ashok Kumar 1. 2. 3. Introduction /-divergence and /„-divergence 79 82 2.1 2.2 2.3 84 84 85 Extension to infinite X Bregman vs Csiszár Classical vs quantum CR inequality Information geometry from a divergence function 86 3.1 3.2 3.3 88 90 95 Information geometry for a-CR inequality An а-version of Cramér֊Rao inequality Generalized version of Cramér-Rao inequality 4. Information geometry for Bayesian CR inequality and Barankin bound 98 5. Information geometry for Bayesian a-CR inequality 101 6. Information geometry for Hybrid CR inequality 106 7. Summary 106 Acknowledgments 107 Appendix 107 A.1 Other generalizations of Cramér-Rao inequality References Section II Theoretical applications and physics 6. 107 110 115 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 117 B. Roy Frieden 1. Introduction 1.1 On learning, energy, sensory messages 1.2 On variational approaches 1.3 Vital role played by information 118 119 119 2. Overview and comparisons of applications 2.1 2.2 2.3 2.4 2.5 2.6 Classical dynamics Quantum physics Biology Thermodynamics Extending use of the principle ofnatural selection From biological cell to earth to solar system, galaxy, universe, and multiverse 2.7 Creation of a multiverse by requiring its Fisher I to be maximized 2.8 Analogy of a cancer "universe" 2.9 What ultimately causes a multiverse to form? 2.10 Is there empirical evidence for
amultiverse having formed? 118 120 120 120 121 121 122 123 124 125 125 126
viii Contents 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 Details of the process of growing successive universes How many universes N might exist in the multiverse? Annihilation of universes Growth of a bubble of nothing Counter-growth of new universes Possibility of many annihilation waves How large a number N of universes exist? Is the multiverse merely a theoretical construct? Should the fact that we do not, and have not observed life elsewhere in our universe affect a belief that we exist in a multiverse? 3. Derivation of principle of maximum Fisher information (MFI) 3.1 Cramer-Rao (C-R) inequality 3.2 On derivation of the C-R inequality 3.3 What do such data values (augmented by knowledge of a single equality obeyed by the system physics) have to say about the unknown system physics? 4. Kantian view of Fisher information use to predict a physical law 4.1 How principle of maximum information originates with Kant 4.2 On significance of the information difference l—J 5. Principle of minimum loss of Fisher information 5.1 Verifying that minimum loss is actually achieved by the principle 5.2 Summary and foundations of the Fisher approach to knowledge acquisition 5.3 What is accomplished by use of the Fisher approach 6. Commonality of information-based growths of cancer and viral infections 6.1 MFI applied to early cancer growth 6.2 Later-stage cancer growth 6.3 MFI applied to early covid-19 growth 6.4 Common biological causes of cancer- and covid-19 growth; the ACE2 link References 7. Quantum metrology and quantum correlations 126 128 129 129 130 130 130 131 131 132 132
132 133 135 135 135 136 137 137 140 142 142 143 143 144 146 149 Diego G. Bussandri and Pedro W. Lamberti 1. 2. 3. 4. 5. Quantum correlations Parameter estimation Cramer-Rao bound Quantum Fisher information Quantum correlations in estimation theory 5.1 Heisenberg limit 5.2 Interferometric power 6. Conclusion References 149 152 153 155 156 157 159 160 160
Contents 8. IX Information, economics, and the Cramér-Rao bound 161 Raymond J. Hawkins and B. Roy Frieden 1. 2. 3. Introduction Shannon entropy and Fisher information Financial economics 161 162 164 3.1 3.2 164 168 Discount factors and bonds Derivative securities 4. Macroeconomics 5. Discussion and summary Acknowledgments References 9. Zipfs law results from the scaling invariance of the Cramer-Rao inequality 171 174 175 175 179 Alberto Hernando and Angelo Plastino 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 179 180 180 180 181 183 183 183 Section 111 Advanced statistical theory 10. Я-Deformed probability families with subtractive and divisive normalizations 185 187 /un Zhang and Ting-Kam Leonard Wong 1. 2. 3. Introduction 187 1.1 1.2 1.3 189 191 192 Deformation models Deformed probability families: General approach Chapter outline A-Deformation of exponential and mixture families 193 2.1 2.2 2.3 2.4 2.5 193 194 195 196 197 A-Deformation Deformation: Subtractive approach Deformation: Divisive approach Relation between the two normalizations А-Exponential and A-mixture families Deforming Legendre duality: A-Duality 199 3.1 3.2 3.3 199 201 3.4 From Bregman divergence to A-logarithmic divergence А-Deformed Legendre duality Relationship between А-conjugation and Legendre conjugation Information geometry of A-logarithmic divergence 202 206
x Contents 4. A-Deformedentropy and divergence 4.1 Relation between potential functions and Rényi entropy 4.2 Relation between A-Iogarithmic divergence and Rényi divergence 4.3 Entropy maximizing property of A-exponential family 5. Example: A-Deformation of the probability simplex 5.1 А-Exponential representation 5.2 A-Mixture representation 6. Summary and conclusion References 11. 207 208 209 210 210 211 212 214 Some remarks on Fisher information, the Cramer-Rao inequality, andtheir applicationsto physics 217 H.C. Miller, Angelo Plastino, and A.R. Plastino I. Introduction 2. Diffusion equation 3. Connection with Tsallis statistics 4. Conclusions Appendix A.1 The Cramer-Rao bound References Index 207 217 220 222 225 226 226 227 229 |
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