by: Prasanta S. Bandyopadhyay, Malcolm R. Forster
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Statisticians and philosophers of science have many common interests but restricted communication with each other. This volume aims to remedy these shortcomings. It provides state-of-the-art research in the area of Philosophy of Statistics by encouraging numerous experts to communicate with one another without feeling "restricted" by their disciplines or thinking "piecemeal" in their treatment of issues. A second goal of this book is to present work in the field without bias toward any particular statistical paradigm.Broadly speaking, the essays in this Handbook are concerned with problems of induction, statistics and probability. For centuries, foundational problems like induction have been among philosophers' favorite topics; recently, however, non-philosophers have increasingly taken a keen interest in these issues. This volume accordingly contains papers by both philosophers and non-philosophers, including scholars from nine academic disciplines.Provides a bridge between philosophy and current scientific findingsCovers theory and applicationsEncourages multi-disciplinary dialogue
Contents
Philosophy of Statistics: An Iintroduction
1 Philosophy, Statistics, and Philosophy of Statistics
2 Four Statistical Paradigms
3 The Likelihood Principle
4 The Curve-Fitting Problem, Problem of Induction, and Role of Simplicity in Inference
5 Recent Advances in Model Selection
6 Causal Inference in Observational Studies
7 Specific Topics of Interest
8 An Application of Statistics to Climate Science
9 Historical Topics in Probability and Statistics
10 Plans Behind Arranging Major Sections/Chapters
11 CODA
Acknowledgements
Bibliography
Part I: Probability & Statistics
Elementary Probability and Statistics: A Primer
1 Introduction
3 From Probablity to Statistics and a Road-Map for the Rest of the Chapter
4 Data Represented and Described
5 Random Variables and Probability Distributions
6 Statistical Inference
7 Conclusion
Acknowledgements
Bibliography
Part II: Philosophical Controversies about Conditional Probability
Conditional Probability
1 Introduction
2 Mathematical Theory
3 Philosophical Applications
4 Problems with the Ratio Analysis of Conditional Probability
5 Kolmorogov's Refinement: Conditional Probability as a Random Variable
6 Conditional Probability as Primitive
7 Conditional Probabilities and Updating Rules
8 Some Paradoxes and Puzzles Involving Conditional Probability and Conditionalization
9 Probabilities of Conditionals and Conditional Probabilities
10 Conclusion
Bibliography
The Varieties of Conditional Probability
1 Pluralism About Conditional Probability
2 Conditional Probability vs. Probability Given the Background
Bibliography
Part III: Four Paradigms of Statistics
Classical Statistics Paradigm
Error Statistics
1 What is Error Statistics?
2 A Philosophy for Error Statistics
3 Error Statistics vs. The Likelihood Principle
4 Error Statistics is Self-Correcting: Testing Statistical Model Assumptions
Bibliography
Significance Testing
1 Introduction
2 The Development and Logic of Significance Tests
3 (Mis)Interpreting Significance Tests
4 Objections to Significance Tests
5 Conclusion
Acknowledgements
Bibliography
Bayesian Paradigm
The Bayesian Decision-Theoretic Approach to Statistics
1 Bayesianism in Inference and Decision
2 Decision Problems
3 Degrees of Belief and Desire
4 Probability Axioms
5 Conditionalization
6 Bayesian and Classical Statistics Compared
7 Appendix: Savage's Representation Theorem
Bibliography
Modern Bayesian Inference: Foundations and Objective Methods
1 Introduction
2 Foundations
3 The Bayesian Paradigm
4 Reference Analysis
5 Inference Summaries
6 Discussion
Bibliography
Evidential Probability and Objective Bayesian Epistemology
1 Introduction
2 Evidential Probability
3 Second-Order Evidential Probability
4 Objective Bayesian Epistemology
5 EP-Calibrated Objective Bayesianism
6 Conclusion
Acknowledgements
Bibliography
Confirmation Theory
Introduction
1 The Probabilistic Logic of Confirmation Functions
2 Two Conceptions of Confirmational Probability
3 The Logical Structure of Evidential Support and the Role of Bayes' Theorem in that Logic
4 What is Confirmational Probability Anyway?
5 The Likelihood Ratio Convergence Theorem
6 When the Likelihoods are Vague AND/OR Diverse
Acknowledgements
Bibliography
Challenges to Bayesian Confirmation Theory
1 Introduction
2 Competing Accounts of the Nature of Inductive Inference
3 Challenges to Framework Assumptions
4 Additivity
5 Bayesian Dynamics
6 Further Challenges
7 Conclusion
Acknowledgements
Bibliography
Bayesianism as a Pure Logic of Inference
1 Introduction
2 Bolzano and Partial Entailment
3 Symmetry and its Problems
4 Carnap's Logical Probability
5 From Logical Probability to Probabilistic Logic
6 Conditional Probability
7 Countable Versus Finite Additivity
8 Conclusion
Acknowledgements
Bibliography
Bayesian Inductive Logic, Verisimilitude, and Statistics
1 Bayesian Inductive Logic and Bayesian Statistics
2 Theory of Inductive Probabilities, Confirmation, and Statistics
3 Verisimilitude and Statistics
Acknowledgements
Bibliography
Likelihood Paradigm
Likelihood and its Evidential Framework
1 Introduction
2 The Likelihood Paradigm
3 Reexamination of Accumulating Data ('multiple Looks')
4 Measuring Evidence about Several Endpoints Simultaneously ('multiple Comparisons')
5 Comments
Bibliography
Evidence, Evidence Functions, and Error Probabilities
1 Introduction
2 Quantifying Evidence, Likelihood Ratios and Evidence Functions
3 The Probability of Misleading Evidence and Inference Reliability
4 Global & Local Reliability
5 Local Reliability and the Evidential Paradigm
6 Evidence and Composite Hypotheses
7 Selecting Between Composite Hypotheses
8 Evidence and the Challenges of Multiplicities
9 Discussion
Acknowledgements
Bibliography
Akaikean Paradigm
AIC Scores as Evidence: A Bayesian Interpretation
1 Introduction
2 Estimates as Evidence
3 Differences in AIC Scores
4 Conclusion
Appendix
Acknowledgements
Bibliography
Part IV: The Likelihood Principle
The Likelihood Principle
1 Introduction
2 Calculation, Justification and Classification
3 Terminology and Caveats
4 The Likelihood Principle: Precise Statement
5 A Proof of the Likelihood Principle
6 A Proof of the Likelihood Principle: Continuation from the WSP and the WCP
7 Other Versions of the Likelihood Principle
8 Is the Likelihood Function Well Defined?
Acknowledgements
Bibliography
Part V: Recent Advances in Model Selection
AIC, BIC and Recent Advances in Model Selection
Overview
1 Examples
2 The Akaike Information Criterion (AIC)
3 Bayes Factor and BIC
4 Comparison of AIC and BIC Through an Example
5 Recent Advances in Model Selection
Summing Up
Acknowledgements
Bibliography
Posterior Model Probabilities
Introduction
1 Marginal Likelihood
2 Asymptotics
3 Improper Priors
4 Subjective Model Weights
5 'Fully Objective' Formal Priors
6 Problems with the Jeffreys Measure
7 Examples
8 Conclusion
Acknowledgements
Bibliography
A Appendix: Proof of Theorem 1
Part VI: Attempts to Understand Different Aspects of "Randomness”
Defining Randomness
Randomness Finally Defined
Inadequacies and Deficiencies
Commonalities
Acknowledgements
Bibliography
Mathematical Foundations of Randomness
1 Introduction
2 Strings, Sequences, Cantor Space, and Lebesgue Measure
3 Classical Stochastic Randomness in Infinite Sequences
4 Algorithms and Post Machines
5 Von Mises' Definition of Random Sequence
6 Martin-Löf and Solovay Randomness
7 Randomness of Finite Strings: Kolmogorov Complexity
8 The Prefix-Free Complexity K
9 Kolmogorov-Chaitin Randomness and Schnorr's Theorem
10 Relative and Stronger Randomness. Hierarchies
11 Randomness via Martingales. Other Frequentist Definitions
12 Conclusion. The Martin-Löf-Chaitin Thesis
Acknowledgements
Bibliography
Part VII: Probabilistic and Statistical Paradoxes
Paradoxes of Probability
1 Introduction
2 Puzzles and Paradoxes
3 Simple Puzzles of Probability
4 More Complex Problems of Probability
5 Paradoxes of Probability and Decision
Acknowledgements
Bibliography
Statistical Paradoxes: Take it to the Limit
1 Introduction
2 Frequentist vs. Bayesian
3 Lindley's Paradox
4 Fieller-Creasy Problem
5 Concluding Remark
Acknowledgements
Bibliography
Part VIII: Statistics and Inductive Inference
Statistics as Inductive Inference
1 Statistical Procedures as Inductive Logics
2 Observational Data
3 Inductive Inference
4 Neyman-Pearson Testing
5 Fisher's Parameter Estimation
6 Carnapian Logics
7 Bayesian Statistics
8 Bayesian Inductive Logic
9 Neyman-Pearson Test as an Inference
10 In Conclusion
Acknowledgements
Bibliography
Part IX: Various Issues about Causal Inference
Common Cause in Causal Inference
1 Introduction
2 Structural Equation Models
3 Conditioning Versus Manipulating
4 Estimating Manipulated Means
5 Open Problems
6 Appendix
Acknowledgements
Bibliography
The Logic and Philosophy of Causal Inference: A Statistical Perspective
Do we need Philosophy of Causation for a Statistical Theory of Causal Inference?
Potential Outcomes and Structural Equations
Causal Systems and Causal Diagrams
Discussion
Acknowledgements
Bibliography
Part X: Some Philosophical Issues Concerning Statistical Learning Theory
Statistical Learning Theory as a Framework for the Philosophy of Induction
Pattern Recognition
Bayes Error Rate R*
Using Data to Learn the Statistical Probability Distribution?
Empirical Risk Minimization
Data Coverage Balanced Against Something Else
Philosophical Implications
Conclusion
Bibliography
Testability and Statistical Learning Theory
VC Dimension
Acknowledgements
Bibliography
Part XI: Different Approaches to Simplicity Related to Inference and Truth
Luckiness and Regret in Minimum Description Length Inference
1 Introduction
2 Encoding the Hypothesis: A Matter of Luckiness and Regret
3 Encoding the Data
4 The Purpose of MDL
5 MDL in Perspective
Bibliography
MML, Hybrid Bayesian Network Graphical Models, Statistical Consistency, Invariance and Uniqueness
1 Introduction
2 Information Theory — And Varieties Thereof
3 Probabilistic Inference, Log-Loss Scoring and Kullback-Leibler Distance — And Uniqueness
4 Ockham's Razor (And Misunderstandings) And MML
5 Desiderata: Statistical Invariance, Statistical Consistency, Efficiency, Small-Sample Performance, Etc.
6 Minimum Message Length (MML) and Strict MML
7 MML and Some Applications in Philosophy and Elsewhere
8 Acknowledgements
Bibliography
Simplicity, Truth, and Probability
1 Introduction
2 The Argument from Bayes Factors
3 The Argument from Over-Fitting
4 Ockham's Causal Razor
5 Efficient Pursuit of the Truth
6 Empirical Simplicity Defined
7 Inquiry and Ockham's Razor
8 A Basic Ockham Efficiency Theorem
9 Stability, Errors and Retraction Times
10 Extension to Branching Simplicity
11 When Defeat does not Imply Refutation
12 Extension to Randomized Scientific Strategies
13 Disjunctive Beliefs, Retraction Degrees, and a Gettier Example
14 Extension to Degrees of Belief
15 Conclusion
Acknowledgements
Bibliography
Part XII: Special Problems in Statistics/Computer Science
Normal Approximations
1 Introduction
2 The Central Limit Theorem
3 The Delta Method and Slutsky's Theorem
4 Discrete Distributions
5 Randomization Inference
6 Likelihood-Based Inference
7 Bayesian Posterior Distributions
8 Expansions
A Selected Mathematical and Statistical Concepts
B Heuristic Proofs of Selected Theorems
C Regularity Conditions
Acknowledgements
Bibliography
Stein's Phenomenon
1 Introduction
2 Stein's Insight
3 A Data Analysis Example
4 A Geometric Heuristic
5 An Empirical Bayes Perspective
6 A Regression Perspective
7 Related Methodological Developments
8 Stein's Phenomenon and Modern Science
Acknowledgements
Bibliography
Data, Data, Everywhere: Statistical Issues in Data Mining
1 Oceans of Data
2 Knowledge Discovery from Data
3 Monkeys and Typewriters; Bangladeshi Butter and the S&P 500
4 Underfitting and Overfitting
5 Testing and Evaluation
6 Assumptions and Violations
7 The Case of Associations in Association Rules
8 Replacing Statisticians With Computers
Bibliography
Part XIII: An Application of Statistics to Climate Change
An Application of Statistics in Climate Change: Detection of Nonlinear Changes in a Streamflow Timing Measure in the Columbia and Missouri Headwaters
1 Introduction
2 Previous Research
3 Data
4 Trend Detection Methods
5 Additive Mixed Models
6 Results
7 Conclusion
Bibliography
Part XIV: Historical Approaches to Probability/Statistics
The Subjective and the Objective
1 The Era of Good Feelings
2 Liberté. Egalité, Probabilité
3 Antoine Augustin Cournot
4 The Influence of Kant
5 1842 and 1843: The Anni Mirabili
6 De Morgan vs. Venn: "And there has not yet been Philosophy Enough to Expel it ..."
7 Bertrand: Objectivity as Consensus
8 Poincaré: The Method of Arbitrary Functions
9 Conclusion
Bibliography
Probability in Ancient India
Introduction: Mathematical Pre-Requisites
The Notion of a Fair Game and the Frequentist Interpretation of Probability
Subjective Probabilities and the Underlying Logic of Sentences
Index
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