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The theory of probability, based on very general rules referred to as the Cox-Polya-Jaynes Desiderata, can be used both as a theory of random mass phenomena and as a quantitative theory of plausible inference about the parameters of…
Bayesian probability theory is used as a framework to develop a formalism for the scientific method based on principles of inductive reasoning. The formalism allows for precise definitions of the key concepts in theories of physics and also…
In this paper we study the interaction between logic and probability. In particular, we show that the convex hull of evaluations of a broad class of logics is always effectively axiomatizable. We define a Birkhoff-style calculus for…
The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to…
We review briefly the concepts underlying complex systems and probability distributions. The later are often taken as the first quantitative characteristics of complex systems, allowing one to detect the possible occurrence of regularities…
In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. We introduce a…
We show that the quantum-mechanical probability distribution involving complex probability amplitudes can be derived from three natural conditions imposed on a relativistically invariant probability function describing the motion of a…
As observers of the universe we are physical systems within it. If the universe is very large in space and/or time, the probability becomes significant that the data on which we base predictions is replicated at other locations in…
This study has the purpose of addressing four questions that lie at the base of the probability theory and statistics, and includes two main steps. As first, we conduct the textual analysis of the most significant works written by eminent…
We survey the Kolmogorov's approach to the notion of randomness through the Kolmogorov complexity theory. The original motivation of Kolmogorov was to give up a quantitative definition of information. In this theory, an object is randomness…
An explicit bound is given for the Kolmogorov distance between a mixture of normal distributions and a normal distribution with properly chosen parameter values. A random variable X has a mixture of normal distributions if its conditional…
The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's…
A number of well-known theorems, such as Cox's theorem and de Finetti's theorem. prove that any model of reasoning with uncertain information that satisfies specified conditions of "rationality" must satisfy the axioms of probability…
This paper presents an approach for developing the explanation capabilities of rule-based expert systems managing imprecise and uncertain knowledge. The treatment of uncertainty takes place in the framework of possibility theory where the…
We formalize the idea of probability distributions that lead to reliable predictions about some, but not all aspects of a domain. The resulting notion of `safety' provides a fresh perspective on foundational issues in statistics, providing…
It is shown that the standard Kolmogorov model for probability spaces cannot in general allow the elimination but of only a small amount of probabilistic redundancy. This issue, a purely theoretical weakness, not necessarily related to…
The purpose of this paper is to present a mathematical theory that can be used as a foundation for statistics that include improper priors. This theory includes improper laws in the initial axioms and has in particular Bayes theorem as a…
The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra.…
Probability measures by themselves, are known to be inappropriate for modeling the dynamics of plain belief and their excessively strong measurability constraints make them unsuitable for some representational tasks, e.g. in the context of…
This paper addresses the central question of what a coherent concept of probability might look like that would do justice to both classical probability theory, axiomatized by Kolmogorov, and quantum theory. At a time when quanta are…