A Mathematical Theory of Evidence

Glenn Shafer

Princeton University Press, Apr 21, 1976 - Business & Economics - 297 pages

Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his "lower probabilities" as epistemic probabilities and taking his rule for combining "upper and lower probabilities" as fundamental.


The book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called "monotone of order of infinity." and Dempster's rule for combining such set functions. This rule, together with the idea of "weights of evidence," leads to both an extensive new theory and a better understanding of the Bayesian theory. The book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem.

Contents

• INTRODUCTION: 3
• 1 Synopsis: 4
• 2 The Idea of Chance: 9
• 3 The Doctrine of Chances: 12
• 4 Chances as Degrees of Belief: 16
• 5 The Bayesian Theory of Partial Belief: 18
• 6 The Role of Judgment: 20
• 7 The Representation of Ignorance: 22
• 8 Combination vs Conditioning: 25
• 9 The Representation of Probable Reasoning: 28
• 10 Statistical Inference: 29
• 11 The Bayesian Theory as a Limiting Case: 32
• 12 Probability: 33
• DEGREES OF BELIEF: 35
• 2 Basic Probability Numbers: 37
• 3 Belief Functions: 39
• 4 Commonality Numbers: 40
• 5 Degrees of Doubt and Upper Probabilities: 42
• 6 Bayesian Belief Functions: 44
• 7 Mathematical Appendix: 46
• DEMPSTERS RULE OF COMBINATION: 57
• 1 Combining Two Belief Functions: 58
• 2 Multiplying Commonality Numbers: 61
• 3 Combining Several Belief Functions: 62
• 4 The Weight of Conflict: 64
• 5 Conditioning Belief Functions: 66
• 6 Other Properties of Dempsters Rule: 67
• 7 Mathematical Appendix: 68
• SIMPLE AND SEPARABLE SUPPORT FUNCTIONS: 74
• 1 Simple Support Functions: 75
• 3 The Weight of Evidence: 77
• 4 Heterogeneous Evidence: 79
• 5 Conflicting Evidence: 82
• 6 Separable Support Functions: 86
• THE WEIGHTS OF EVIDENCE: 88
• 1 Decomposing Separable Support Functions: 89
• 2 Combining Weights of Evidence: 90
• 3 The Assessment of Evidence: 93
• 4 The Weight of Internal Conflict: 95
• 5 The Impingement Function: 97
• 6 The WeightofConflict Conjecture: 98
• 7 Some Numerical Examples: 100
• 8 Mathematical Appendix: 103
• COMPATIELE FRAMES OF DISCERNMENT 1 Refinements and Coarsenings: 115
• 2 The Inner and Outer Reductions: 117
• 3 Is There an U1timate Refinement?: 119
• 4 Families of Compatible Frames: 121
• 5 Consistent Belief Functions: 125
• 6 Independent Frames: 127
• 7 Mathematical Appendix: 129
• SUPPORT FUNCTIONS: 141
• 1 The Class of Support Functions: 142
• 2 Support Dubiety and Plausibility: 144
• 3 The Vacuous Extension of a Support Function: 146
• 4 Evidential Independence: 147
• 5 Cognitive Independence: 149
• 6 Mathematical Appendix: 152
• THE DISCERNMENT OF EVIDENCE: 172
• 1 Families of Compatible Support Functions: 173
• 2 Discerning the Interaction of Evidence: 175
• 3 Discerning Weights of Evidence: 182
• 4 If the WeightofConflict Conjecture is True: 185
• 5 Mathematical Appendix: 186
• CHAPTER 9 QUASI SUPPORT FUNCTIONS: 196
• 1 Infinite Contradictory Evidence: 197
• 2 The Class of Quasi Support Functions: 199
• 3 Chances are not Degrees of Support: 201
• 4 The Bayesian Profusion of Infinite Weights: 202
• 5 Bayes Theorem: 204
• 6 Mathematical Appendix: 209
• CONSONANCE: 219
• 2 The Contour Function: 222
• 3 The Embarrassment of Dissonance: 223
• 4 Inferential Evidence: 226
• 5 Mathematical Appendix: 229
• STATISTICAL EVIDENCE: 237
• 1 A Convention for Assessing Statistical Evidence: 238
• 2 The Weights of Evidence: 245
• 3 Epistemic vs Aleatory Combination: 247
• 4 The Effect of a Bayesian Prior: 250
• 5 Discounting Statistical Evidence: 251
• 6 Specifications on Compatible Frames: 255
• 7 The Role of Supposition: 258
• 8 Perspective: 262
• 9 Mathematical Appendix: 264
• THE DUAL NATURE OF PROBABLE REASONING: 274
• 2 The Need for Assumptions: 276
• 3 Choosing our Frames of Discernment: 279
• 4 The Role of Epistemic Probability: 284
• 5 Two Tasks: 285
• BIBLIOGRAPHY: 287
• INDEX: 292

About the author

Glenn Shafer is professor at Rutgers University and director of the Ph.D. program.

Bibliographic information

• Title: A Mathematical Theory of Evidence; Volume 42 of Limited paperback editions
• Author: Glenn Shafer
• Edition: illustrated, reprint
• Publisher: Princeton University Press, 1976
• ISBN: 069110042X, 9780691100425
• Length: 297 pages
• Subjects: Business & Economics / Finance / General; Mathematics / History & Philosophy; Mathematics / Probability & Statistics / Bayesian Analysis; Mathematics / Probability & Statistics / General; Philosophy / Epistemology