Interpretations of Probability

Contents:

Introduction

I. FOUNDATIONS OF PROBABILITY THEORY

A few words about measures

Classical and ensemble definitions of probability

Frequency theory of probability

Kolmogorov=s measure-theoretical approach

Kolmogorov=s ideas on probability

Measure-theoretical approach and interpretations of probability

Subjective (Bayesian) probability theory

Foundations of randomness

Operation of combining of collectives

Independence of collectives

Frequency and measure-theoretical viewpoints on independence

Generalization of the operation of combining

Comparative probability

II. QUANTUM PROBABILITIES

Classical and quantum probability rules

Physical interpretations of the wave function

`Contradiction' between quantum and classical probability calculi

Probabilities with respect to objective conditions

Einstein-Poldolsky-Rosen paradox: Probability, reality, locality

Bell's inequality for probabilities

Bell's mystification

Bell's inequality for covariations

Hidden variables and Bell's inequality

III. NEGATIVE PROBABILITIES

The origin of negative probabilities in the ensemble and frequency theories

Signed `probabilistic' measures and Einstein-Podolsky-Rosen paradox

Wiegener phase-space distribution and negative probability

Dirac's world with negative probabilities

Negative probabilities and localization

IV. P-ADIC PROBABILITY THEORY

Non-Archimedean number-systems; p-adic numbers

Frequency probability theory

Ensemble probability

Measures

p-Adic probability space

V. INFORMATION AND PROBABILITY

Information reality

Dynamics on information spaces

Information velocity, acceleration, mass and force, Newton's laws

Mathematical `pathologies'

Quantum mechanics on information spaces

The pilot wave theory for cognitive systems

Subjective probability as probability with respect to the statistical ensemble of human ideas

Bayes' theorem and Freud's psychoanalysis

VI. TESTS FOR RANDOMNESS FOR P-ADIC PROBABILITY THEORY

p-Adic probability measures on the space of binary sequences

Some technical p-adic results

p-Adic tests for randomness

Limit theorems

Recursive enumeration of the set of p-adic tests

No p-adic universal test

Randomness of infinite sequences

Bibliography

Index

1999; iv pages
ISBN 90-6764-310-6
Price: EUR 123/US$ 143