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Measuring Statistical Evidence Using Relative Belief [Kõva köide]

(University of Toronto, Ontario, Canada)
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Measuring Statistical Evidence Using Relative BeliefSuurem pilt
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Püsilink: https://www.kriso.ee/db/9781482242799.html
Märksõnad:
Neglecting to define how to measure evidence is a significant failure for any proposed theory of statistical inference, declares Evans. He summarizes recent research on developing a theory of statistical inference that is based on measuring statistical evidence, and shows how being explicit about how to measure statistical evidence addresses the basic question of when a statistical analysis is correct. His topics are statistical problems, probability, characterizing statistical evidence, measuring statistical evidence using relative belief, and choosing and checking the model and prior. Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

A Sound Basis for the Theory of Statistical Inference

Measuring Statistical Evidence Using Relative Belief provides an overview of recent work on developing a theory of statistical inference based on measuring statistical evidence. It shows that being explicit about how to measure statistical evidence allows you to answer the basic question of when a statistical analysis is correct.

The book attempts to establish a gold standard for how a statistical analysis should proceed. It first introduces basic features of the overall approach, such as the roles of subjectivity, objectivity, infinity, and utility in statistical analyses. It next discusses the meaning of probability and the various positions taken on probability. The author then focuses on the definition of statistical evidence and how it should be measured. He presents a method for measuring statistical evidence and develops a theory of inference based on this method. He also discusses how statisticians should choose the ingredients for a statistical problem and how these choices are to be checked for their relevance in an application.

Arvustused

"This creative book by Michael Evans describes not only a new way to measure the strength of evidence but also a system of statistical data analysis on the basis of that measure. That system, specifically designed in response to problems with conventional frequentist and Bayesian statistics, has advantages over previous solutions to those problems. The core of the system is its concept of the degree of relative belief in a hypothesis. The author defines relative belief in a hypothesis of positive prior probability as the ratio of the posterior probability that the hypothesis is true to the prior probability that it is true. The relative belief in a simple null hypothesis of 0 prior probability is then defined as the relative belief in a hypothesis of positive prior probability in the limit as it shrinks toward the simple hypothesis. In short, Evans makes concrete recommendations for evidential data analysis based on informed reflection on Lindley's paradox, the logical coherence principle, and the conditional probability principle." David R. Bickel, in Mathematical Reviews Clippings, December 2017

"This book presents a comprehensive survey of the literature on measuring statistical evidence based on relative beliefsThis book integrates recent developments into a coursebook format with a pedagogical introduction to the area. The author is a main contributor in the field and the book systematically assembles material conveying his findings and his views on statistics as a wholeThe first three chapters provide the readers with a basic introduction to the foundations of probability and statistics viewed through the lens of the authors school of thoughtThe second part of the book is devoted to the measurement of statistical evidence based on relative beliefsThough the first (overview) part of the book does not presuppose much background knowledge, my feeling is that a sound grasp of the subject will help the reader come away with a solid understanding and interpretation of this tour. On the other hand, the main theoretical part does not require many technical prerequisites and is accessible to students having a basic background." Markus Bibinger, University of Marburg, in Journal of the American Statistical Association, Volume 111, 2016

" a fascinating book, in that it defines a coherent and original approach to the vexing question of assessing and measuring evidence in a statistical problem. And spells out most vividly the issue of prior checking. the book is a great opportunity to discuss this approach and to oppose it to potential alternatives, hopefully generating incoming papers and talks." Christian Robert on his blog Xi'ans Og, July 2015

"The book is a pleasure to read and presents a really unique view of statistics as a whole and how to measure evidence in particular. this is one of the most important books written recently on the foundations of statistics, providing a modern and logical perspective on the reasons for good Bayesian statistical practice." David Nott, National University of Singapore

Preface xv
1 Statistical Problems 1(26)
1.1 Introduction
1(1)
1.2 Statistical Problems
2(2)
1.3 Statistical Models
4(2)
1.4 Infinity and Continuity in Statistics
6(4)
1.5 The Principle of Empirical Criticism
10(4)
1.5.1 The Objectivity of the Data
12(1)
1.5.2 The Subjectivity of Statistical Models
13(1)
1.5.3 The Subjective Prior
14(1)
1.6 The Concept of Utility
14(2)
1.7 The Principle of Frequentism
16(2)
1.8 Statistical Inferences
18(1)
1.9 Example
19(7)
1.9.1 Checking the Model
20(1)
1.9.2 Checking for Prior-Data Conflict
21(2)
1.9.3 Statistical Inference
23(2)
1.9.4 Checking the Prior for Bias
25(1)
1.10 Concluding Comments
26(1)
2 Probability 27(24)
2.1 Introduction
27(4)
2.1.1 Kolmogorov Axioms
27(1)
2.1.2 Conditional Probability
28(3)
2.2 Principle of Insufficient Reason
31(4)
2.3 Subjective Probability
35(12)
2.3.1 Comparative or Qualitative Probability
35(2)
2.3.2 Probability via Betting
37(3)
2.3.3 Probability and No Arbitrage
40(3)
2.3.4 Scoring Rules
43(1)
2.3.5 Savage's Axioms
44(2)
2.3.6 Cox's Theorem
46(1)
2.4 Relative Frequency Probability
47(3)
2.4.1 Long-Run Relative Frequency
48(1)
2.4.2 Randomness
49(1)
2.5 Concluding Comments
50(1)
3 Characterizing Statistical Evidence 51(44)
3.1 Introduction
51(1)
3.2 Pure Likelihood Inference
51(7)
3.2.1 Inferences for the Full Parameter
51(4)
3.2.2 Inferences for a Marginal Parameter
55(2)
3.2.3 Prediction Problems
57(1)
3.2.4 Summarizing the Pure Likelihood Approach
58(1)
3.3 Sufficiency, Ancillarity and Completeness
58(8)
3.3.1 The Sufficiency Principle
59(2)
3.3.2 The Conditionality Principle
61(3)
3.3.3 Birnbaum's Theorem
64(2)
3.3.4 Completeness
66(1)
3.4 p-Values and Confidence
66(5)
3.4.1 p-Values and Tests of Significance
66(2)
3.4.2 Neyman—Pearson Tests
68(1)
3.4.3 Rejection Trials and Confidence Regions
69(2)
3.4.4 Summarizing the Frequentist Approach
71(1)
3.5 Bayesian Inferences
71(19)
3.5.1 Basic Concepts
72(5)
3.5.2 Likelihood, Sufficiency and Conditionality
77(1)
3.5.3 MAP-Based Inferences
78(3)
3.5.4 Quantile-Based Inferences
81(1)
3.5.5 Loss-Based Inferences
82(1)
3.5.6 Bayes Factors
83(5)
3.5.7 Hierarchical Bayes
88(1)
3.5.8 Empirical Bayes
88(1)
3.5.9 Bayesian Frequentism
89(1)
3.5.10 Summarizing the Bayesian Approach
90(1)
3.6 Fiducial Inference
90(3)
3.7 Concluding Comments
93(2)
4 Measuring Statistical Evidence Using Relative Belief 95(72)
4.1 Introduction
95(1)
4.2 Relative Belief Ratios and Evidence
96(10)
4.2.1 Basic Definition of a Relative Belief Ratio
97(5)
4.2.2 General Definition of a Relative Belief Ratio
102(4)
4.3 Other Proposed Measures of Evidence
106(7)
4.3.1 The Bayes Factor
108(2)
4.3.2 Good's Information and Weight of Evidence
110(1)
4.3.3 Desiderata for a Measure of Evidence
111(2)
4.4 Measuring the Strength of the Evidence
113(5)
4.4.1 The Strength of the Evidence
114(4)
4.5 Inference Based on Relative Belief Ratios
118(11)
4.5.1 Hypothesis Assessment
119(2)
4.5.2 Estimation
121(2)
4.5.3 Prediction Inferences
123(1)
4.5.4 Examples
123(6)
4.6 Measuring the Bias in the Evidence
129(6)
4.7 Properties of Relative Belief Inferences
135(23)
4.7.1 Consistency
135(4)
4.7.2 Convergence of Bias Measures
139(1)
4.7.3 Optimality of Relative Belief Credible Regions
140(4)
4.7.4 Optimality of Relative Belief Hypothesis Assessment
144(2)
4.7.5 Optimality of Relative Belief Estimation
146(7)
4.7.5.1 Finite ψ
147(2)
4.7.5.2 Countable ψ
149(1)
4.7.5.3 General ψ
150(2)
4.7.5.4 Relative Belief Credible Regions and Loss
152(1)
4.7.6 Robustness of Relative Belief Inferences
153(5)
4.8 Concluding Comments
158(1)
4.9 Appendix
159(8)
4.9.1 Proof of Proposition 4.7.5
159(11)
4.9.1.1 Proof of Proposition 4.7.9
160(1)
4.9.1.2 Proof of Proposition 4.7.12
161(1)
4.9.1.3 Proof of Proposition 4.7.13
162(1)
4.9.1.4 Proof of Proposition 4.7.14 and Corollary 4.7.10
162(1)
4.9.1.5 Proof of Proposition 4.7.16
163(1)
4.9.1.6 Proof of Proposition 4.7.17
164(1)
4.9.1.7 Proof of Proposition 4.7.18
164(1)
4.9.1.8 Proof of Lemma 4.7.1
165(1)
4.9.1.9 Proof of Proposition 4.7.19
166(1)
5 Choosing and Checking the Model and Prior 167(44)
5.1 Introduction
167(1)
5.2 Choosing the Model
168(2)
5.3 Choosing the Prior
170(6)
5.3.1 Eliciting Proper Priors
170(2)
5.3.2 Improper Priors
172(4)
5.4 Checking the Ingredients
176(2)
5.5 Checking the Model
178(9)
5.5.1 Checking a Single Distribution
179(2)
5.5.2 Checks Based on Minimal Sufficiency
181(4)
5.5.3 Checks Based on Ancillaries
185(2)
5.6 Checking the Prior
187(13)
5.6.1 Prior-Data Conflict
188(4)
5.6.2 Prior-Data Conflict and Ancillaries
192(2)
5.6.3 Hierarchical Priors
194(2)
5.6.4 Hierarchical Models
196(2)
5.6.5 Invariant p-Values for Checking for Prior-Data Conflict
198(1)
5.6.6 Diagnostics for the Effect of Prior-Data Conflict
199(1)
5.7 Modifying the Prior
200(9)
5.8 Concluding Comments
209(2)
6 Conclusions 211(4)
A The Definition of Density 215(4)
A.1 Defining Densities
215(1)
A.2 General Spaces
216(3)
References 219(10)
Index 229
Michael Evans is a professor in the Department of Statistics at the University of Toronto. His research focuses on statistical inference, particularly a theory of inference based on the concept of relative belief. He is an associate editor of Bayesian Analysis and a former president of the Statistical Society of Canada.