The 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.
1. 1 Introduction - p. 16 2. 2 Pythagoras theorems in information geometry - p. 18 3. 3 Power entropy and divergence - p. 24 4. 4 Linear regression model - p. 31 5. 5 Generalized linear model - p. 35 6. 6 Discussion - p. 40 7. References - p. 41 8. Further reading - p. 42 9. Chapter 3: Rao distances and conformal mapping - p. 43 10. Chapter 4: Cramer-Rao inequality for testing the suitability of divergent partition functions - p. 57 11. Chapter 5: Information geometry and classical Cramér–Rao-type inequalities - p. 79 12. Abstract - p. 79 13. 1 Introduction - p. 79 14. 2 I-divergence and Iα-divergence - p. 82 15. 3 Information geometry from a divergence function - p. 86 16. 4 Information geometry for Bayesian CR inequality and Barankin bound - p. 97 17. 5 Information geometry for Bayesian α-CR inequality - p. 101 18. 6 Information geometry for Hybrid CR inequality - p. 106 19. 7 Summary - p. 106 20. References - p. 108 21. 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 - p. 117 22. Chapter 7: Quantum metrology and quantum correlations - p. 149 23. Chapter 8: Information, economics, and the Cramér-Rao bound - p. 161 24. Chapter 9: Zipf's law results from the scaling invariance of the Cramer–Rao inequality - p. 179 25. Chapter 10: λ-Deformed probability families with subtractive and divisive normalizations - p. 187 26. Abstract - p. 187 27. 1 Introduction - p. 187 28. 2 λ-Deformation of exponential and mixture families - p. 193 29. 3 Deforming Legendre duality: λ-Duality - p. 199 30. 4 λ-Deformed entropy and divergence - p. 207 31. 5 Example: λ-Deformation of the probability simplex - p. 210 32. 6 Summary and conclusion - p. 214 33. References - p. 213 34. Chapter 11: Some remarks on Fisher information, the Cramer–Rao inequality, and their applications to physics - p. 217 35. Abstract - p. 217 36. 1 Introduction - p. 217 37. 2 Diffusion equation - p. 219 38. 3 Connection with Tsallis statistics - p. 221 39. 4 Conclusions - p. 225 40. References - p. 226 41. Index - p. 228
The 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.
1. 1 Introduction - p. 16 2. 2 Pythagoras theorems in information geometry - p. 18 3. 3 Power entropy and divergence - p. 24 4. 4 Linear regression model - p. 31 5. 5 Generalized linear model - p. 35 6. 6 Discussion - p. 40 7. References - p. 41 8. Further reading - p. 42 9. Chapter 3: Rao distances and conformal mapping - p. 43 10. Chapter 4: Cramer-Rao inequality for testing the suitability of divergent partition functions - p. 57 11. Chapter 5: Information geometry and classical Cramér–Rao-type inequalities - p. 79 12. Abstract - p. 79 13. 1 Introduction - p. 79 14. 2 I-divergence and Iα-divergence - p. 82 15. 3 Information geometry from a divergence function - p. 86 16. 4 Information geometry for Bayesian CR inequality and Barankin bound - p. 97 17. 5 Information geometry for Bayesian α-CR inequality - p. 101 18. 6 Information geometry for Hybrid CR inequality - p. 106 19. 7 Summary - p. 106 20. References - p. 108 21. 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 - p. 117 22. Chapter 7: Quantum metrology and quantum correlations - p. 149 23. Chapter 8: Information, economics, and the Cramér-Rao bound - p. 161 24. Chapter 9: Zipf's law results from the scaling invariance of the Cramer–Rao inequality - p. 179 25. Chapter 10: λ-Deformed probability families with subtractive and divisive normalizations - p. 187 26. Abstract - p. 187 27. 1 Introduction - p. 187 28. 2 λ-Deformation of exponential and mixture families - p. 193 29. 3 Deforming Legendre duality: λ-Duality - p. 199 30. 4 λ-Deformed entropy and divergence - p. 207 31. 5 Example: λ-Deformation of the probability simplex - p. 210 32. 6 Summary and conclusion - p. 214 33. References - p. 213 34. Chapter 11: Some remarks on Fisher information, the Cramer–Rao inequality, and their applications to physics - p. 217 35. Abstract - p. 217 36. 1 Introduction - p. 217 37. 2 Diffusion equation - p. 219 38. 3 Connection with Tsallis statistics - p. 221 39. 4 Conclusions - p. 225 40. References - p. 226 41. Index - p. 228
