Elements of Bayesian statistics /
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| Author / Creator: | Florens, J. P. |
|---|---|
| Imprint: | New York : M. Dekker, c1990. |
| Description: | xxxi, 499 p. : ill. ; 24 cm. |
| Language: | English |
| Series: | Monographs and textbooks in pure and applied mathematics 134 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/1068857 |
Table of Contents:
- Preface
- Notation
- 0.. Basic Tools and Notation from Probability Theory
- 0.1.. Introduction
- 0.2.. Measurable Spaces
- 0.2.1.. [sigma]-Fields
- 0.2.2.. Measurable Functions
- 0.2.3.. Product of Measurable Spaces
- 0.2.4.. Monotone Class Theorems
- 0.3.. Probability Spaces
- 0.3.1.. Measures and Integrals
- 0.3.2.. Probabilities. Expectations. Null Sets
- 0.3.3.. Transition and Product Probability
- 0.3.4.. Conditional Expectation
- 0.3.5.. Densities
- 1.. Bayesian Experiments
- 1.1.. Introduction
- 1.2.. The Basic Concepts of Bayesian Experiments
- 1.2.1.. General Definitions
- 1.2.2.. Dominated Experiments
- 1.2.3.. Three Remarks on Regular and Dominated Experiments
- 1.2.4.. A Remark Regarding the Interpretation of Bayesian Experiments
- 1.2.5.. A Remark on Sampling Theory and Bayesian Methods
- 1.2.6.. A Remark Regarding So-called "Improper" Prior Distributions
- 1.2.7.. Families of Bayesian Experiments
- 1.3.. Some Examples of Bayesian Experiments
- 1.4.. Reduction of Bayesian Experiments
- 1.4.1.. Introduction
- 1.4.2.. Marginal Experiments
- 1.4.3.. Conditional Experiment
- 1.4.4.. Complementary Reductions
- 1.4.5.. Dominance in Reduced Experiments
- 2.. Admissible Reductions: Sufficiency and Ancillarity
- 2.1.. Introduction
- 2.2.. Conditional Independence
- 2.2.1.. Notation
- 2.2.2.. Definition of Conditional Independence
- 2.2.3.. Null Sets and Completion
- 2.2.4.. Basic Properties of Conditional Independence
- 2.2.5.. Conditional Independence and Densities
- 2.2.6.. Conditional Independence as Point Properties
- 2.3.. Admissible Reductions of an Unreduced Experiment
- 2.3.1.. Introduction
- 2.3.2.. Admissible Reductions on the Sample Space
- 2.3.3.. Admissible Reductions on the Parameter Space
- 2.3.4.. Some Comments on the Definitions
- 2.3.5.. Elementary Properties of Sufficiency and Ancillarity
- 2.3.6.. Sufficiency and Ancillarity in a Dominated Experiment
- 2.3.7.. Sampling Theory and Bayesian Methods
- 2.3.8.. A First Result on the Relations between Sufficiency and Ancillarity
- 3.. Admissible Reductions in Reduced Experiments
- 3.1.. Introduction
- 3.2.. Admissible Reduction in Marginal Experiments
- 3.2.1.. Introduction
- 3.2.2.. Basic Concepts
- 3.2.3.. Sufficiency and Ancillarity in Unreduced and in Marginal Experiments
- 3.2.4.. A Remark on "Partial" Sufficiency
- 3.3.. Admissible Reductions in Conditional Experiments
- 3.3.1.. Introduction
- 3.3.2.. Reductions in the Sample Space
- 3.3.3.. Reductions in the Parameter Space
- 3.3.4.. Elementary Properties
- 3.3.5.. Relationships between Sufficiency and Ancillarity
- 3.3.6.. Sufficiency and Ancillarity in a Dominated Reduced Experiment
- 3.4.. Jointly Admissible Reductions
- 3.4.1.. Mutual Sufficiency
- 3.4.2.. Mutual Exogeneity
- 3.4.3.. Bayesian Cut
- 3.4.4.. Joint Reductions in a Dominated Experiment
- 3.4.5.. Joint Reductions in a Conditional Experiment
- 3.4.6.. Some Examples
- 3.5.. Comparison of Experiments
- 3.5.1.. Comparison on the Sample Space: Sufficiency
- 3.5.2.. Comparison on the Parameter Space: Encompassing
- 4.. Optimal Reductions: Maximal Ancillarity and Minimal Sufficiency
- 4.1.. Introduction
- 4.2.. Maximal Ancillarity
- 4.3.. Projections of [sigma]-Fields
- 4.3.1.. Introduction
- 4.3.2.. Definition and Elementary Properties
- 4.3.3.. Projections and Conditional Independence
- 4.4.. Minimal Sufficiency
- 4.4.1.. Minimal Sufficiency in Unreduced and in Marginal Experiments
- 4.4.2.. Elementary Properties of Minimal Sufficiency
- 4.4.3.. Minimal Sufficiency in a Dominated Experiment
- 4.4.4.. Sampling Theory and Bayesian Methods
- 4.4.5.. Minimal Sufficiency in Conditional Experiment
- 4.4.6.. Optimal Mutual Sufficiency
- 4.5.. Identification Among [sigma]-Fields
- 4.6.. Identification in Bayesian Experiments
- 4.6.1.. Identification in a Reduced Experiment
- 4.6.2.. Sampling Theory and Bayesian Methods
- 4.7.. Exact and Totally Informative Experiments
- 4.8.. Punctual Exact Estimability
- 5.. Optimal Reductions: Further Results
- 5.1.. Introduction
- 5.2.. Measurable Separability
- 5.3.. Measurable Separability in Bayesian Experiments
- 5.3.1.. Measurably Separated Bayesian Experiment
- 5.3.2.. Basu Second Theorem
- 5.3.3.. Sampling Theory and Bayesian Methods
- 5.4.. Strong Identification of [sigma]-Fields
- 5.4.1.. Definition and General Properties
- 5.4.2.. Strong Identification and Conditional Independence
- 5.4.3.. Minimal Splitting
- 5.5.. Completeness in Bayesian Experiments
- 5.5.1.. Completeness and Sufficiency
- 5.5.2.. Completeness and Ancillarity
- 5.5.3.. Successive Reductions of a Bayesian Experiment
- 5.5.4.. Sampling Theory and Bayesian Methods
- 5.5.5.. Identifiability of Mixtures
- 6.. Sequential Experiments
- 6.1.. Introduction
- 6.2.. Sequences of Conditional Independences
- 6.3.. Sequential Experiments
- 6.3.1.. Definition of Sequential Experiments
- 6.3.2.. Admissible Reductions in Sequential Experiments
- 6.4.. Transitivity
- 6.4.1.. Basic Theory
- 6.4.2.. Markovian Property and Transitivity
- 6.5.. Relations Among Admissible Reductions
- 6.5.1.. Admissible Reductions on the Parameter Space
- 6.5.2.. Admissible Reductions on the Sample Space
- 6.5.3.. Admissible Reductions in Joint Reductions
- 6.6.. The Role of Transitivity: Further Results
- 6.6.1.. Weakening of Transitivity Conditions
- 6.6.2.. Necessity of Transitivity Conditions
- 7.. Asymptotic Experiments
- 7.1.. Introduction
- 7.2.. Limit of Sequences of Conditional Independences
- 7.3.. Asymptotically Admissible Reductions
- 7.3.1.. Asymptotic Properties of Sequential Experiments
- 7.3.2.. Asymptotic Sufficiency
- 7.3.3.. Asymptotic Admissibility of Joint Reductions
- 7.3.4.. Asymptotically Admissible Reductions in Conditional Experiments
- 7.4.. Asymptotic Exact Estimability
- 7.4.1.. Exact Estimability and Bayesian Consistency
- 7.4.2.. Sampling Theory and Bayesian Methods
- 7.5.. Estimability of Discrete [sigma]-Fields
- 7.6.. Mutual Conditional Independence and Conditional 0-1 Laws
- 7.6.1.. Mutual Conditional Independence
- 7.6.2.. Sifted Sequences of [sigma]-Fields
- 7.7.. Tail-Sufficient and Independent Bayesian Experiments
- 7.7.1.. Bayesian Tail-Sufficiency
- 7.7.2.. Bayesian Independence
- 7.7.3.. Independent Tail-Sufficient Bayesian Experiments
- 7.8.. An Example
- 7.8.1.. Global and Sequential Analysis
- 7.8.2.. Asymptotic Analysis
- 7.8.3.. The Case [beta] = [infinity]
- 7.8.4.. The Case [beta less than sign infinity]
- 8.. Invariant Experiments
- 8.1.. Introduction
- 8.2.. Invariance, Ergodicity and Mixing
- 8.2.1.. Invariant Sets and Functions
- 8.2.2.. Invariance as Point Properties
- 8.2.3.. Invariance and Conditional Invariance of [sigma]-Fields
- 8.2.4.. Ergodicity and Mixing
- 8.2.5.. Existence of Invariant Measure
- 8.2.6.. Randomization of the Set of Transformations
- 8.3.. Invariant Experiments
- 8.3.1.. Construction and Definition of an Invariant Bayesian Experiment
- 8.3.2.. Invariance and Reduction
- 8.3.3.. Invariance and Exact Estimability
- 9.. Invariance in Stochastic Processes
- 9.1.. Introduction
- 9.2.. Bayesian Stochastic Processes and Representations
- 9.2.1.. Introduction
- 9.2.2.. Representation of Experiments
- 9.2.3.. Bayesian Stochastic Processes
- 9.2.4.. Shift and Permutations
- 9.3.. Standard Bayesian Stochastic Processes
- 9.3.1.. Stationary Processes
- 9.3.2.. Exchangeable and i.i.d. Processes
- 9.3.3.. Moving Average Processes
- 9.3.4.. Markovian Stationary Processes
- 9.3.5.. Autoregressive Moving Average Processes
- 9.3.6.. An Example
- 9.4.. Conditional Stochastic Processes
- 9.4.1.. Introduction
- 9.4.2.. Shift in Conditional Stochastic Processes
- 9.4.3.. Conditional Shift-Invariance
- Bibliography
- Author Index
- Subject Index
