Anyone involved in the philosophy of science is naturally drawn into the study of the foundations of probability. Different interpretations of probability, based on competing philosophical ideas, lead to different statistical techniques, and frequently to mutually contradictory consequences.This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries. Mathematical models are constructed based on this interpretation, and statistical inference and decision theory are applied, including some examples in artificial intelligence, solving the main foundational problems. Nonstandard analysis is extensively developed for the construction of the models and in some of the proofs. Many nonstandard theorems are proved, some of them new, in particular, a representation theorem that asserts that any stochastic process can be approximated by a process defined over a space with equiprobable outcomes.
Author(s): Rolando Chuaqui (Eds.)
Series: North-Holland Mathematics Studies 166
Edition: 1
Publisher: Elsevier, Academic Press
Year: 1991
Language: English
Pages: ii-xix, 1-484
Content:
Editor
Page ii
Edited by
Page iii
Copyright page
Page iv
Dedication
Page ivA
Preface
Pages v-viii
List of Figures
Page xix
Preliminaries
Pages 1-14
Chapter I The Problem of Foundations
Pages 17-40
Chapter II Truth, Possibility, and Probability
Pages 41-55
Chapter III Probability Models
Pages 57-84
Chapter IV The Basic Principles of Inference and Decision
Pages 85-116
Chapter V A Medical Example
Pages 117-125
Chapter VI Introduction to Probability Theory
Pages 129-157
Chapter VI Disjunctive Probability Spaces and Invariant Measures
Pages 159-170
Chapter VIII Elements of Infinitesimal Analysis
Pages 171-195
Chapter IX Integration
Pages 197-226
Chapter X Probability Distributions
Pages 227-251
Chapter XI Hyperfinite Random Processes
Pages 253-276
Chapter XII Simple Probability Structures
Pages 279-291
Chapter XIII The structure of chance
Pages 293-317
Chapter XIV Equiprobability structures
Pages 319-341
Chapter XV Probability structures
Pages 343-351
Chapter XVI Examples of probability structures
Pages 353-364
Chapter XVII Logical probability
Pages 365-376
Chapter XVIII Classical statistical inference
Pages 379-411
Chapter XIX Problems in statistical inference
Pages 413-430
Chapter XX Decision theory
Pages 431-442
Appendix A Foundations of nonstandard analysis
Pages 443-459
Appendix B Extensions of integrals and measures
Pages 461-468
Bibliography
Pages 469-473
Index
Pages 474-484
