Probabilistic Events and Physical Reality: A Complete Algebra of Probability

Probability theory can be modified in essentially one way while maintaining consistency with the basic Bayesian framework. This modification results in copies of standard probability theory for real, complex or quaternion probabilities. These copies, in turn, allow one to derive quantum theory while restoring standard probability theory in the classical limit. The argument leading to these three copies constrain physical theories in the same sense that Cox's original argument...

This is the logical foundation for for Relativity Theory, Probability Theory, and for Quantum Theory. Contents is the following: 1 Introduction. 2 Classical logic. 3 Time and space. 3.1 Recorders. 3.2 Time. 3.3 Space. 3.4 Relativity. 4. Probability. 4.1 B-functions. 4.2 Independent tests. 4.3 Function of probability. 4.4 Conditional probability. 4.5 Classical probability 4.6 B-functions and classical propositional logic. 4.7 Consistency of the probability function. 4.7.1 No...

This book introduces to the theory of probabilities from the beginning. Assuming that the reader possesses the normal mathematical level acquired at the end of the secondary school, we aim to equip him with a solid basis in probability theory. The theory is preceded by a general chapter on counting methods. Then, the theory of probabilities is presented in a discrete framework. Two objectives are sought. The first is to give the reader the ability to solve a large number of p...

Probabilistic models require the notion of event space for defining a probability measure. An event space has a probability measure which ensues the Kolmogorov axioms. However, the probabilities observed from distinct sources, such as that of relevance of documents, may not admit a single event space thus causing some issues. In this article, some results are introduced for ensuring whether the observed prob- abilities of relevance of documents admit a single event space. Mor...

The aim of the article is to argue that the interpretations of quantum mechanics and of probability are much closer than usually thought. Indeed, a detailed analysis of the concept of probability (within the standard frequency theory of R. von Mises) reveals that the latter concept always refers to an observing system. The enigmatic role of the observer in the Copenhagen interpretation therefore derives from a precise understanding of probability. Besides explaining several e...

This is the first of the proposed sets of notes to be published in the website Gonit Sora (http://gonitsora.com). The notes will hopefully be able to help the students to learn their subject in an easy and comprehensible way. These notes are aimed at mimicking exactly what would be typically taught in a one-semester course at a college or university. The level of the notes would be roughly at the undergraduate level. The present sets of notes are not yet complete and this is ...

Andrei Kolmogorov's Grundbegriffe der Wahrscheinlichkeits-rechnung put probability's modern mathematical formalism in place. It also provided a philosophy of probability--an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe--the work of the earlier scholars whose ideas Kolmogorov synthesized.

This text presents an unified approach of probability and statistics in the pursuit of understanding and computation of randomness in engineering or physical or social system with prediction with generalizability. Starting from elementary probability and theory of distributions, the material progresses towards conceptual and advances in prediction and generalization in statistical models and large sample theory. We also pay special attention to unified derivation approach and...

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an "equality" such that its two sides are unequal probabilities. The deduced expressions are erroneous only when it can be prov...

This is an introduction to some of the most probabilistic aspects of free probability theory.