Elements of Bayesian statistics:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
New York, NY u.a.
Dekker
1990
|
| Edition: | 1. print. |
| Series: | Pure and applied mathematics
134. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 463 - 487 |
| Physical Description: | XXXI, 499 S. |
| ISBN: | 0824781236 |
Staff View
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Record in the Search Index
| _version_ | 1804122914768814080 |
|---|---|
| adam_text | Contents
Preface v
Notation xxv
0. Basic Tools and Notation from Probability Theory 1
0.1. Introduction 1
0.2. Measurable Spaces 2
0.2.1. r Fields 2
0.2.2. Measurable Functions 5
0.2.3. Product of Measurable Spaces 9
0.2.4. Monotone Class Theorems 11
0.3. Probability Spaces 13
xv
xvi Contents
0.3.1. Measures and Integrals 13
0.3.2. Probabilities. Expectations. Null Sets 15
0.3.3. Transition and Product Probability 17
0.3.4. Conditional Expectation 18
0.3.5. Densities 22
1. Bayesian Experiments 25
1.1. Introduction 25
1.2. The Basic Concepts of Bayesian Experiments 26
1.2.1. General Definitions 26
1.2.2. Dominated Experiments 28
1.2.3. Three Remarks on Regular and Dominated
Experiments 31
1.2.4. A Remark Regarding the Interpretation of Bayesian
Experiments 32
1.2.5. A Remark on Sampling Theory and Bayesian Methods 33
1.2.6. A Remark Regarding So called Improper Prior
Distributions 34
1.2.7. Families of Bayesian Experiments 35
1.3. Some Examples of Bayesian Experiments 36
1.4. Reduction of Bayesian Experiments 46
1.4.1. Introduction 46
1.4.2. Marginal Experiments 47
1.4.3. Conditional Experiment 51
1.4.4. Complementary Reductions 55
1.4.5. Dominance in Reduced Experiments 59
Contents xvii
2. Admissible Reductions: Sufficiency and Ancillarity 65
2.1. Introduction 65
2.2. Conditional Independence 66
2.2.1. Notation 66
2.2.2. Definition of Conditional Independence 67
2.2.3. Null Sets and Completion 69
2.2.4. Basic Properties of Conditional Independence 71
2.2.5. Conditional Independence and Densities 73
2.2.6. Conditional Independence as Point Properties 76
2.3. Admissible Reductions of an Unreduced Experiment 77
2.3.1. Introduction 77
2.3.2. Admissible Reductions on the Sample Space 78
2.3.3. Admissible Reductions on the Parameter Space 79
2.3.4. Some Comments on the Definitions 79
2.3.5. Elementary Properties of Sufficiency and Ancillarity 83
2.3.6. Sufficiency and Ancillarity in a Dominated Experiment 84
2.3.7. Sampling Theory and Bayesian Methods 87
2.3.8. A First Result on the Relations between Sufficiency
and Ancillarity 91
3. Admissible Reductions in Reduced Experiments 95
3.1. Introduction 95
3.2. Admissible Reduction in Marginal Experiments 99
3.2.1. Introduction 99
3.2.2. Basic Concepts 99
3.2.3. Sufficiency and Ancillarity in Unreduced and in
Marginal Experiments 100
xviii Contents
3.2.4. A Remark on Partial Sufficiency 102
3.3. Admissible Reductions in Conditional Experiments 105
3.3.1. Introduction 105
3.3.2. Reductions in the Sample Space 106
3.3.3. Reductions in the Parameter Space 107
3.3.4. Elementary Properties 108
3.3.5. Relationships between Sufficiency and Ancillarity 109
3.3.6. Sufficiency and Ancillarity in a Dominated Reduced
Experiment 111
3.4. Jointly Admissible Reductions 111
3.4.1. Mutual Sufficiency 112
3.4.2. Mutual Exogeneity 116
3.4.3. Bayesian Cut 118
3.4.4. Joint Reductions in a Dominated Experiment 120
3.4.5. Joint Reductions in a Conditional Experiment 122
3.4.6. Some Examples 125
3.5. Comparison of Experiments 128
3.5.1. Comparison on the Sample Space: Sufficiency 128
3.5.2. Comparison on the Parameter Space: Encompassing 134
4. Optimal Reductions: Maximal Ancillarity and
Minimal Sufficiency 139
4.1. Introduction 139
4.2. Maximal Ancillarity 140
4.3. Projections of cr Fields 143
4.3.1. Introduction 143
4.3.2. Definition and Elementary Properties 143
Contents xix
4.3.3. Projections and Conditional Independence 145
4.4. Minimal Sufficiency 153
4.4.1. Minimal Sufficiency in Unreduced and in
Marginal Experiments 153
4.4.2. Elementary Properties of Minimal Sufficiency 154
4.4.3. Minimal Sufficiency in a Dominated Experiment 155
4.4.4. Sampling Theory and Bayesian Methods 156
4.4.5. Minimal Sufficiency in Conditional Experiment 157
4.4.6. Optimal Mutual Sufficiency 158
4.5. Identification Among r Fields 160
4.6. Identification in Bayesian Experiments 165
4.6.1. Identification in a Reduced Experiment 165
4.6.2. Sampling Theory and Bayesian Methods 171
4.7. Exact and Totally Informative Experiments 176
4.8. Punctual Exact Estimability 184
5. Optimal Reductions: Further Results 187
5.1. Introduction 187
5.2. Measurable Separability 189
5.3. Measurable Separability in Bayesian Experiments 197
5.3.1. Measurably Separated Bayesian Experiment 197
5.3.2. Basu Second Theorem 200
5.3.3. Sampling Theory and Bayesian Methods 202
5.4. Strong Identification of r Fields 203
5.4.1. Definition and General Properties 204
5.4.2. Strong Identification and Conditional
Independence 208
xx Contents
5.4.3. Minimal Splitting 214
5.5. Completeness in Bayesian Experiments 222
5.5.1. Completeness and Sufficiency 222
5.5.2. Completeness and Ancillarity 226
5.5.3. Successive Reductions of a Bayesian
Experiment 230
5.5.4. Sampling Theory and Bayesian Methods 235
5.5.5. Identifiability of Mixtures 238
6. Sequential Experiments 241
6.1. Introduction 241
6.2. Sequences of Conditional Independences 242
6.3. Sequential Experiments 247
6.3.1. Definition of Sequential Experiments 247
6.3.2. Admissible Reductions in Sequential
Experiments 249
6.4. Transitivity 253
6.4.1. Basic Theory 253
6.4.2. Markovian Property and Transitivity 258
6.5. Relations Among Admissible Reductions 267
6.5.1. Admissible Reductions on the Parameter Space 267
6.5.2. Admissible Reductions on the Sample Space 267
6.5.3. Admissible Reductions in Joint Reductions 269
6.6. The Role of Transitivity: Further Results 276
6.6.1. Weakening of Transitivity Conditions 276
6.6.2. Necessity of Transitivity Conditions 280
Contents xxi
7. Asymptotic Experiments 283
7.1. Introduction 283
7.2. Limit of Sequences of Conditional Independences 285
7.3. Asymptotically Admissible Reductions 292
7.3.1. Asymptotic Properties of Sequential
Experiments 292
7.3.2. Asymptotic Sufficiency 293
7.3.3. Asymptotic Admissibility of Joint Reductions 297
7.3.4. Asymptotically Admissible Reductions in
Conditional Experiments 301
7.4. Asymptotic Exact Estimability 306
7.4.1. Exact Estimability and Bayesian Consistency 306
7.4.2. Sampling Theory and Bayesian Methods 307
7.5. Estimability of Discrete r Fields 313
7.6. Mutual Conditional Independence
and Conditional 0 1 Laws 318
7.6.1. Mutual Conditional Independence 319
7.6.2. Sifted Sequences of © Fields 324
7.7. Tail Sufficient and Independent
Bayesian Experiments 329
7.7.1. Bayesian Tail Sufficiency 329
7.7.2. Bayesian Independence 333
7.7.3. Independent Tail Sufficient Bayesian
Experiments 335
7.8. An Example 339
7.8.1. Global and Sequential Analysis 340
7.8.2. Asymptotic Analysis 342
7.8.3. The Case /? = oo 345
xxii Contents
7.8.4. The Case /3 oo 345
8. Invariant Experiments 349
8.1. Introduction 349
8.2. Invariance, Ergodicity and Mixing 352
8.2.1. Invariant Sets and Functions 352
8.2.2. Invariance as Point Properties 355
8.2.3. Invariance and Conditional Invariance of
(r Fields 359
8.2.4. Ergodicity and Mixing 372
8.2.5. Existence of Invariant Measure 373
8.2.6. Randomization of the Set of Transformations 376
8.3. Invariant Experiments 381
8.3.1. Construction and Definition of
an Invariant Bayesian Experiment 381
8.3.2. Invariance and Reduction 390
8.3.3. Invariance and Exact Estimability 400
9. Invariance in Stochastic Processes 403
9.1. Introduction 403
9.2. Bayesian Stochastic Processes and Representations 404
9.2.1. Introduction 404
9.2.2. Representation of Experiments 405
9.2.3. Bayesian Stochastic Processes 408
9.2.4. Shift and Permutations 410
9.3. Standard Bayesian Stochastic Processes 415
9.3.1. Stationary Processes 415
Contents xxiii
9.3.2. Exchangeable and i.i.d. Processes 420
9.3.3. Moving Average Processes 429
9.3.4. Markovian Stationary Processes 430
9.3.5. Autoregressive Moving Average Processes 437
9.3.6. An Example 440
9.4. Conditional Stochastic Processes 448
9.4.1. Introduction 448
9.4.2. Shift in Conditional Stochastic Processes 451
9.4.3. Conditional Shift Invariance 452
Bibliography 463
Author Index 489
Subject Index 493
|
| any_adam_object | 1 |
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| author_facet | Florens, Jean P. Mouchart, Michel Rolin, Jean-Marie |
| author_role | aut aut aut |
| author_sort | Florens, Jean P. |
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| building | Verbundindex |
| bvnumber | BV008585258 |
| classification_rvk | SK 830 |
| ctrlnum | (OCoLC)246624931 (DE-599)BVBBV008585258 |
| discipline | Mathematik |
| edition | 1. print. |
| format | Book |
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| spelling | Florens, Jean P. Verfasser aut Elements of Bayesian statistics Jean Pierre Florens ; Michel Mouchart ; Jean-Marie Rolin 1. print. New York, NY u.a. Dekker 1990 XXXI, 499 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 134. Literaturverz. S. 463 - 487 Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd rswk-swf CD-ROM (DE-588)4139307-7 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 s DE-604 Bayes-Entscheidungstheorie (DE-588)4144220-9 s CD-ROM (DE-588)4139307-7 s 1\p DE-604 Mouchart, Michel Verfasser aut Rolin, Jean-Marie Verfasser aut Pure and applied mathematics 134. (DE-604)BV000001885 134 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005641911&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Florens, Jean P. Mouchart, Michel Rolin, Jean-Marie Elements of Bayesian statistics Pure and applied mathematics Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd CD-ROM (DE-588)4139307-7 gnd Bayes-Verfahren (DE-588)4204326-8 gnd |
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| title | Elements of Bayesian statistics |
| title_auth | Elements of Bayesian statistics |
| title_exact_search | Elements of Bayesian statistics |
| title_full | Elements of Bayesian statistics Jean Pierre Florens ; Michel Mouchart ; Jean-Marie Rolin |
| title_fullStr | Elements of Bayesian statistics Jean Pierre Florens ; Michel Mouchart ; Jean-Marie Rolin |
| title_full_unstemmed | Elements of Bayesian statistics Jean Pierre Florens ; Michel Mouchart ; Jean-Marie Rolin |
| title_short | Elements of Bayesian statistics |
| title_sort | elements of bayesian statistics |
| topic | Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd CD-ROM (DE-588)4139307-7 gnd Bayes-Verfahren (DE-588)4204326-8 gnd |
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