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

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