Reflections on the Foundations of Probability and Statistics: Essays in Honor of Teddy Seidenfeld 54

This Festschrift celebrates Teddy Seidenfeld and his seminal contributions to philosophy, statistics, probability, game theory and related areas. The 13 contributions in this volume, written by leading researchers in these fields, are supplemented by an interview with Teddy Seidenfeld that offers an abbreviated intellectual autobiography, touching on topics of timeless interest concerning truth and uncertainty. Indeed, as the eminent philosopher Isaac Levi writes in this volume: "In a world dominated by Alternative Facts and Fake News, it is hard to believe that many of us have spent our life’s work, as has Teddy Seidenfeld, in discussing truth and uncertainty." The reader is invited to share this celebration of Teddy Seidenfeld’s work uncovering truths about uncertainty and the penetrating insights they offer to our common pursuit of truth in the face of uncertainty. Preface Contents 1 An Interview with Teddy Seidenfeld References 2 The Value Provided by a Scientific Explanation 2.1 Introduction 2.2 Deductive-Nomological [D-N] Explanations 2.2.1 Three Kinds of D-N Explanations 2.2.2 Explanation and Prediction: A Necessary Condition for an Explanation 2.3 Probabilistic Explanations 2.4 Confirming a New Theory with Old Evidence 2.5 Subjunctive Conditionals and Probabilistic Explanations 2.5.1 A Probabilistic Version of Russell's Example 2.5.2 Observations and Interventions 2.5.3 Interventions in the Factory Example 2.5.4 Equilibria and Subjunctives 2.5.5 Counterfactual Conditionals 2.6 Summary References 3 A Gentle Approach to Imprecise Probability 3.1 Bayesianism in a Nutshell 3.2 A Simple Extension 3.3 A Surprising Implication 3.4 Gambles and Events 3.5 Credal Sets and Lower Probability 3.6 Gambles and Lower Previsions 3.7 Acceptable Gambles and Partial Preference Orders 3.8 Coherent Acceptable Gambles 3.9 A Brief Word on Conditional Lower Previsions 3.10 Coherence as a Guide to Life References 4 Foundations For Temporal Reasoning Using Lower Previsions Without A Possibility Space 4.1 Introduction 4.2 Lower Previsions and Desirability 4.3 Future Beliefs as Gambles 4.4 Temporal Sure Preference 4.4.1 The Temporal Sure Preference Principle 4.4.2 Temporal Natural Extension 4.4.3 Further Implications 4.5 Temporal Coherence for Previsions 4.5.1 Basic Structure 4.5.2 Temporal Natural Extension 4.6 Conclusion 4.7 Proofs References 5 On the Equivalence of Normal and Extensive Form Representations of Games 5.1 Introduction 5.2 Equivalence of Extensive Form Games 5.3 On Sufficiency of the Reduced Normal Form 5.4 Reduced Normal Form Representation with Imprecise Probabilities 5.5 Concluding Remarks References 6 Dilation and Informativeness 6.1 Imprecise Probability and Dilation 6.2 What's Weird About Dilation? 6.3 Irrelevance 6.4 Measuring Information with Interval Width 6.5 Measuring Information Through Subsets 6.6 Measuring Information Through Distances 6.7 Measuring Information with Entropy 6.8 Conclusion 6.9 Calculations 6.9.1 Distances 6.9.2 Set-Valued Entropy 6.9.3 Mork's Measure References 7 Playing with Sets of Lexicographic Probabilities and Sets of Desirable Gambles 7.1 Introduction 7.2 Lexicographic Probabilities and Sets of Desirable Gambles 7.2.1 A Bit of Background 7.2.2 Lexicographic Probabilities 7.2.3 Sets of Desirable Gambles 7.2.4 Marginalization and Conditioning 7.2.5 Isomorphism 7.3 Full Conditional Probabilities: Not Really 7.3.1 A Brief Review 7.3.2 Admissibility and Marginalization 7.4 Convexity? 7.5 Non-uniqueness and Weakness 7.6 Discussion References 8 How to Assess Coherent Beliefs: A Comparison of Different Notions of Coherence in Dempster-Shafer Theory of Evidence 8.1 Introduction 8.2 Preliminaries 8.3 Notions of Coherence for Belief Functions 8.4 Special Cases 8.4.1 Finitely Additive Probability Measures 8.4.2 Finitely Minitive Necessity Measures 8.5 Proper Scoring Rules and Correction of an Incoherent Assessment 8.6 Conclusions References 9 Expected Utility in 3D 9.1 Introduction 9.2 State-Dependence 9.3 Act-Dependence 9.4 Conclusion Appendix References 10 On the Normative Status of Mixed Strategies 10.1 Introduction 10.2 Classical Decision Theory 10.3 Non-classical Decision Theories 10.3.1 Imprecise Probability 10.3.2 Incommensurable Values 10.3.3 Inherent Value in the Process of Randomizing 10.3.4 Fairness 10.3.5 Imperfect Recall 10.3.6 Act–State Dependence 10.4 Game Theory 10.4.1 Classical Game Theory 10.4.2 Putting Beliefs into Equilibrium 10.4.3 Zero-Sum Games 10.4.3.1 Being Out-Thought 10.4.3.2 Learning 10.4.4 Can Randomization Be Eliminated? 10.4.4.1 Endogenizing Randomization in Poker 10.4.4.2 Other Ways of Being Unlearnable 10.4.4.3 Harsanyi's Purification Argument 10.5 Regarding RCTs 10.6 Establishing Rules 10.7 Conclusion Dedication References 11 On a Notion of Independence Proposed by Teddy Seidenfeld 11.1 Context and Introduction 11.2 A Crash Course in Desirability-Based Choice Functions 11.2.1 Choice Functions Based on Linear Previsions 11.2.2 Choice Functions Based on Lower Previsions 11.2.3 An Axiomatic Basis for Working with Linear and Lower Previsions 11.3 S-Irrelevance for Events 11.3.1 S-Irrelevance with Respect to Linear Prevision Models 11.3.2 S-Irrelevance with Respect to Lower Prevision Models 11.4 S-Irrelevance for Variables 11.4.1 Defining S-Irrelevance for Variables 11.4.2 S-Irrelevance for Variables with Respect to Lower Prevision Models 11.4.3 S-Irrelevance for Variables with Respect to Linear Prevision Models 11.5 The Far-Reaching Implications of S-Irrelevance and S-Independence References 12 Coherent Choice Functions Without Archimedeanity 12.1 Introduction 12.2 Coherent Choice Functions on Vector Spaces 12.3 The Link with Desirability 12.4 No Representation of Choice Functions on a Binary Space 12.4.1 An Equivalent Characterisation: Rejection Sets 12.4.2 Counterexample 12.5 Weak Archimedeanity 12.6 Discussion Appendix: Proofs References 13 Quantifying Degrees of E-admissibility in Decision Making with Imprecise Probabilities 13.1 Introduction 13.2 The Basic Model 13.2.1 Framework 13.2.2 Criteria for Decision Making 13.3 E-admissibility, Maximality and a Criterion In Between 13.3.1 Comparing E-admissibility and Maximality 13.3.2 The Extents of E-admissible Acts 13.4 The Ordinal Case 13.5 A Stylized Application Example 13.6 Summary and Concluding Remarks References