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.
1. 1 Introduction - p. 22 2. 2 Accelerated optimization - p. 25 3. 3 Hamiltonian-based accelerated sampling - p. 36 4. 4 Statistical inference with kernel-based discrepancies - p. 45 5. 5 Adaptive agents through active inference - p. 54 6. References - p. 65 7. Chapter 3: Equivalence relations and inference for sparse Markov models - p. 79 8. Section II: Information geometry - p. 103 9. Chapter 4: Symplectic theory of heat and information geometry - p. 107 10. Abstract - p. 107 11. 1 Preamble - p. 108 12. 2 Life and seminal work of Souriau on lie groups thermodynamics - p. 118 13. 3 From information geometry to lie groups thermodynamics - p. 117 14. 4 Symplectic structure of fisher metric and entropy as Casimir function in coadjoint representation - p. 123 15. 5 Covariant maximum entropy density by Souriau model - p. 132 16. 6 Conclusion - p. 139 17. References - p. 140 18. Further reading - p. 143 19. Chapter 5: A unifying framework for some directed distances in statistics - p. 145 20. Abstract - p. 145 21. 1 Divergences, statistical motivations, and connections to geometry - p. 146 22. 2 The framework - p. 163 23. 3 Aggregated/integrated divergences - p. 203 24. 4 Dependence expressing divergences - p. 202 25. 5 Bayesian contexts - p. 204 26. 6 Variational representations - p. 207 27. 7 Some further variants - p. 211 28. References - p. 214 29. Chapter 6: The analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions - p. 225 30. Abstract - p. 225 31. 1 Introduction and motivation - p. 225 32. 2 Differential-geometric structures induced by smooth convex functions - p. 227 33. 3 Some illustrating examples - p. 234 34. 4 Information geometry of the mixture family of two distinct Cauchy distributions - p. 240 35. 5 Conclusion - p. 252 36. Appendix Symbolic computing notebook in MAXIMA - p. 252 37. References - p. 255 38. Chapter 7: Local measurements of nonlinear embeddings with information geometry - p. 257 39. Abstract - p. 257 40. 1 Introduction - p. 257 41. 2 α-Divergence and autonormalizing - p. 260 42. 3 α-Discrepancy of an embedding - p. 262 43. 4 Empirical α-discrepancy - p. 267 44. 5 Connections to existing methods - p. 268 45. 6 Conclusion and extensions - p. 271 46. Appendices - p. 274 47. Appendix A Proof of - p. 274 48. Appendix B Proof of - p. 275 49. Appendix C Proof of - p. 276 50. Appendix D Proof of - p. 277 51. References - p. 279 52. Section III: Advanced geometrical intuition - p. 281 53. Chapter 8: Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac motion modeling - p. 285 54. Chapter 9: Geometry and mixture models - p. 327 55. Chapter 10: Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature - p. 357 56. Abstract - p. 357 57. 1 Introduction - p. 357 58. 2 Gaussian distributions and RMT - p. 359 59. 3 Gaussian distributions and Bayesian inference - p. 372 60. Appendix A Riemannian symmetric spaces - p. 383 61. Appendix B Convex optimization - p. 387 62. Appendix C Proofs for - p. 397 63. References - p. 399 64. Chapter 11: Multilevel contours on bundles of complex planes☆ - p. 401 65. Abstract - p. 401 66. 1 Introduction - p. 401 67. 2 Infinitely many bundles of complex planes - p. 402 68. 3 Multilevel contours in a random environment - p. 415 69. 4 Islands and holes in Bℝ(ℂ) - p. 444 70. 5 Concluding remarks - p. 463 71. References - p. 463 72. Index - p. 464
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.
1. 1 Introduction - p. 22 2. 2 Accelerated optimization - p. 25 3. 3 Hamiltonian-based accelerated sampling - p. 36 4. 4 Statistical inference with kernel-based discrepancies - p. 45 5. 5 Adaptive agents through active inference - p. 54 6. References - p. 65 7. Chapter 3: Equivalence relations and inference for sparse Markov models - p. 79 8. Section II: Information geometry - p. 103 9. Chapter 4: Symplectic theory of heat and information geometry - p. 107 10. Abstract - p. 107 11. 1 Preamble - p. 108 12. 2 Life and seminal work of Souriau on lie groups thermodynamics - p. 118 13. 3 From information geometry to lie groups thermodynamics - p. 117 14. 4 Symplectic structure of fisher metric and entropy as Casimir function in coadjoint representation - p. 123 15. 5 Covariant maximum entropy density by Souriau model - p. 132 16. 6 Conclusion - p. 139 17. References - p. 140 18. Further reading - p. 143 19. Chapter 5: A unifying framework for some directed distances in statistics - p. 145 20. Abstract - p. 145 21. 1 Divergences, statistical motivations, and connections to geometry - p. 146 22. 2 The framework - p. 163 23. 3 Aggregated/integrated divergences - p. 203 24. 4 Dependence expressing divergences - p. 202 25. 5 Bayesian contexts - p. 204 26. 6 Variational representations - p. 207 27. 7 Some further variants - p. 211 28. References - p. 214 29. Chapter 6: The analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions - p. 225 30. Abstract - p. 225 31. 1 Introduction and motivation - p. 225 32. 2 Differential-geometric structures induced by smooth convex functions - p. 227 33. 3 Some illustrating examples - p. 234 34. 4 Information geometry of the mixture family of two distinct Cauchy distributions - p. 240 35. 5 Conclusion - p. 252 36. Appendix Symbolic computing notebook in MAXIMA - p. 252 37. References - p. 255 38. Chapter 7: Local measurements of nonlinear embeddings with information geometry - p. 257 39. Abstract - p. 257 40. 1 Introduction - p. 257 41. 2 α-Divergence and autonormalizing - p. 260 42. 3 α-Discrepancy of an embedding - p. 262 43. 4 Empirical α-discrepancy - p. 267 44. 5 Connections to existing methods - p. 268 45. 6 Conclusion and extensions - p. 271 46. Appendices - p. 274 47. Appendix A Proof of - p. 274 48. Appendix B Proof of - p. 275 49. Appendix C Proof of - p. 276 50. Appendix D Proof of - p. 277 51. References - p. 279 52. Section III: Advanced geometrical intuition - p. 281 53. Chapter 8: Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac motion modeling - p. 285 54. Chapter 9: Geometry and mixture models - p. 327 55. Chapter 10: Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature - p. 357 56. Abstract - p. 357 57. 1 Introduction - p. 357 58. 2 Gaussian distributions and RMT - p. 359 59. 3 Gaussian distributions and Bayesian inference - p. 372 60. Appendix A Riemannian symmetric spaces - p. 383 61. Appendix B Convex optimization - p. 387 62. Appendix C Proofs for - p. 397 63. References - p. 399 64. Chapter 11: Multilevel contours on bundles of complex planes☆ - p. 401 65. Abstract - p. 401 66. 1 Introduction - p. 401 67. 2 Infinitely many bundles of complex planes - p. 402 68. 3 Multilevel contours in a random environment - p. 415 69. 4 Islands and holes in Bℝ(ℂ) - p. 444 70. 5 Concluding remarks - p. 463 71. References - p. 463 72. Index - p. 464
