Truth, Possibility and Probability (eBook)

Front Cover 1
Truth, Possibility and Probability 4
Copyright Page 5
Contents 12
Preface 8
List of Figures 20
Preliminaries 22
1. Notation 22
2. Probability 25
Part 1: The concept of probability 36
Chapter I. The problem of foundations 38
1. The uses of probability 38
2. The interpretations of probability 42
3. The Principle of Direct Inference 55
4. Foundations, an a priori inquiry 59
Chapter II. Truth, possibility, and probability 62
1. Truth as related to probability 62
2. Possibility 65
3. Probability based on possibility 70
4. The Principle of Direct Inference 75
Chapter III. Probability models 78
1. Outcomes 78
2. The symmetry or first invariance principle 80
3. The family of events 88
4. The External Invariance Principle 90
5 . Probability models 98
Chapter IV. Principles of inference and decision 106
1. The basis for decisions 107
2. The basis for statistical inference 110
3. Probability and degree of belief 128
4. Probabilities and frequencies 131
Chapter V. A medical example 138
1. The disease model 138
2. The Bayesian model 140
3. The classical statistical inference model 142
Part 2: Elements of Probability Theory 148
Chapter VI. Probability theory 150
1. Probability 150
2. Conditional probability 153
3. Random variables 155
4. Expectation, variance and covariance 157
5. Inequalities 161
6. Algebras of random variables 162
7. Algebras of events 166
8. Conditional distributions and expectations 170
9. Constructions of probability spaces 173
10. Stochastic processes 176
Chapter VII. Disjunctive spaces 180
1. Disjunctive algebras of sets 180
2. Finite disjunctive probability spaces 181
3. Probabilities invariant under groups 185
Chapter VIII. Elements of infinitesimal analysis 192
1. Infinitesimals 194
2. Standard, internal and external sets 196
3. Internal analogues of convergence 204
4. Series 206
5. Convergence of functions 208
6. Continuity 209
7. Differentials and derivatives 210
Chapter IX. Integration 218
1. Hiperfinite sums 218
2. Integrals 224
3. Riemann integrals in higher dimensions 227
4. Transformations 231
5. Curves and surfaces 233
6. Change of variables 237
7. Improper integrals 243
Chapter X. Probability distributions 248
1. Hyperfinite probability spaces 248
2. Discrete distributions 249
3. Continuous distributions 256
4. Normal distributions 264
5. Samples from a normal population 268
Chapter XI. Hyperfinite random processes 274
1. Properties that hold almost everywhere 274
2. The law of large numbers 278
3. Brownian motion and Central Limit Theorem 283
Part 3: Probability models 298
Chapter XII. Simple probability structures 300
1. Relational systems 301
2. The structure of chance 303
3. The probability space 306
4. Simple probability structures 309
Chapter XIII. The structure of chance 314
1. Binary relations and causal relations 315
2. Relational systems with causal structures 319
3. Chance structures 324
4. The group of internal invariance 329
5. Classical structures 336
Chapter XIV. Equiprobability structures 340
1. Equiprobability structures 341
2. Measures on super classical structures 349
3. Homomorphisms 355
4. Induced measures 360
5. The external invariance principle 361
Chapter XV. Probability structures 364
1. Existence of classical structures 366
2. Measure-determined sets 366
3. Probability structures 368
4. Representation theorem 369
Chapter XVI. Examples of probability structures 374
1. Distributions of r balls into n cells 374
2. Selection of a chord at random 377
3. The theory of errors 378
4. Brownian motion 380
5. Models for inference and decision 382
Chapter XVII. Logical probability 386
1. Languages for simple structures 386
2. Languages for compound structures 390
Part 4. Statistical inference and decision theory 398
Chapter XVIII. Classical statistical inference 400
1. General framework 401
2. Discriminating experiments 405
3. Significance tests 411
4. Hypotheses tests 415
5. The Sufficiency Principle 421
6. Point estimation 424
7. Confidence region estimation 427
Chapter XIX. Problems in statistical inference 434
1. The Neyman-Pearson theory 434
2. Alternative hypotheses 438
3. Unsound techniques for estimation 440
4. Likelihood and conditionality 442
5. Initial and final precision 445
6. Gillies’ falsification rule 448
7. Bayesian estimation 449
Chapter XX. Decision theory 452
1. Decision Theory and Consequentialism 452
2. Maximizing expected utility 454
3. Decisions based on evidence 456
4. Decisions in the medical example 459
5. Decision Theory in the law 460
Appendix A. Foundations of nonstandard analysis 464
1. Axioms 465
2. Equivalence of axioms to transfer 472
3. Construction of the nonstandard universe 476
Appendix B. Extensions of integrals and measures 482
1. Internal abstract theory of integration 482
2. Null sets 488
3. Real valued probability measures 489
Bibliography 490
Index 495