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Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative…
Education in statistics, the application of statistics in scientific research, and statistics itself as a scientific discipline are in crisis. Within science, the main cause of the crisis is the insufficiently clarified concept of…
The axiomatic foundation of probability theory presented by Kolmogorov has been the basis of modern theory for probability and statistics. In certain applications it is, however, necessary or convenient to allow improper (unbounded)…
By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a "quantum probability theory". As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum…
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…
We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual…
The crucial but very confidential fact is brought into evidence that, as Kolmogorov himself repeatedly claimed, the mathematical theory of probabilities cannot be applied to physical, factual probabilistic situations because the factual…
The paper aim is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author's main idea was the natural inclusion of Kolmogorov's axiomatics of…
In a Cox model, the partial likelihood, as the product of a series of conditional probabilities, is used to estimate the regression coefficients. In practice, those conditional probabilities are approximated by risk score ratios based on a…
Kolmogorov's setting for probability theory is given an original generalization to account for probabilities arising from Quantum Mechanics. The sample space has a central role in this presentation and random variables, i.e., observables,…
Kolmogorov's foundation of probability takes measure spaces, $\sigma$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum…
Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional…
The probability axioms by R. T. Cox can be regarded as the modern foundations of Bayesian inference, the idea of assigning degrees of belief to logical propositions in a manner consistent with Boolean logic. In this work it is shown that…
According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov's axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the…
Given two events $A$ and $B$, Bayes' law is based on the argument that the probability of $A$ given $B$ is proportional to the probability of $B$ given $A$. When probabilities are interpreted in the Bayesian sense, Bayes' law constitutes a…
There has not been an established mathematical measure of evidence. Some Bayesians have argued that probability can be an objectively correct measure of ``rational degrees of belief,'' which we do not distinguish from evidence. However,…
``Behind every limit theorem, there is an inequality'' said Kolmogorov. We say ``for every inequality, there is an approximate inequality under approximate regularity conditions.'' Suppose $X, X'$ are independent and identically distributed…
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one…
In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a loss function that quantifies how "embarrassing" it…
In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper we show that, despite being prescribed by a fundamental…