scholarly journals Wick polynomials in noncommutative probability: a group-theoretical approach

2021 ◽  
pp. 1-27
Author(s):  
Kurusch Ebrahimi-Fard ◽  
Frédéric Patras ◽  
Nikolas Tapia ◽  
Lorenzo Zambotti
Keyword(s):  
Probability Theory ◽  
Hopf Algebra ◽  
Theoretical Approach ◽  
Algebraic Approach ◽  
Noncommutative Probability ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Wick Products ◽  
Noncommutative Probability Theory

Abstract Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

scholarly journals Upgrading Probability via Fractions of Events

2016 ◽  
Vol 24 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Black And White ◽  
Random Events ◽  
Indicator Functions ◽  
Quantum Phenomena ◽  
Special Case

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

Reasoning About Measurements

2021 ◽  
pp. 31-92
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Physical Theory ◽  
Logical Structure ◽  
Statistical Independence ◽  
Classical Probability ◽  
Multiple Systems ◽  
Classical Probability Theory ◽  
Operational Requirement

This chapter explains the approach of ‘operationalism’, which in a physical theory admits only concepts associated with concrete experimental procedures, and lays out its consequences for propositions about measurements, their logical structure, and states. It illustrates these with toy examples where the ability to perform measurements is limited by design. For systems composed of several constituents this chapter introduces the notions of composite and reduced states, statistical independence, and correlations. It examines what it means for multiple systems to be prepared identically, and how this is represented mathematically. The operational requirement that there must be procedures to measure and prepare a state is examined, and the ensuing constraints derived. It is argued that these constraint leave only one alternative to classical probability theory that is consistent, universal, and fully operational, namely, quantum theory.

scholarly journals Algebras with two multiplications and their cumulants

2019 ◽  
Vol 52 (2) ◽  
pp. 157-186
Author(s):  
Adam Burchardt
Keyword(s):  
Probability Theory ◽  
Linear Space ◽  
Structure Constant ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Sum Of Products ◽  
Probabilistic Concept ◽  
Algebraic Formula

Abstract Cumulants are a notion that comes from the classical probability theory; they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov–Shiryaev’s formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.

Uncertainty about the value of quantum probability for cognitive modeling

2013 ◽  
Vol 36 (3) ◽  
pp. 279-280 ◽  
Author(s):  
Christina Behme
Keyword(s):  
Probability Theory ◽  
Cognitive Modeling ◽  
Quantum Probability ◽  
Convincing Evidence ◽  
Logical Possibility ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Human Rationality

AbstractI argue that the overly simplistic scenarios discussed by Pothos & Busemeyer (P&B) establish at best that quantum probability theory (QPT) is a logical possibility allowing distinct predictions from classical probability theory (CPT). The article fails, however, to provide convincing evidence for the proposal that QPT offers unique insights regarding cognition and the nature of human rationality.

A new rough set based bayesian classifier prior assumption

2020 ◽  
Vol 39 (3) ◽  
pp. 2647-2655
Author(s):  
Naidan Feng ◽  
Yongquan Liang
Keyword(s):  
Probability Theory ◽  
Set Theory ◽  
Rough Set ◽  
Prior Probability ◽  
Rough Set Theory ◽  
Uncertain Data ◽  
Bayesian Classifier ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Public Datasets

 Aiming at the imprecise and uncertain data and knowledge, this paper proposes a novel prior assumption by the rough set theory. The performance of the classical Bayesian classifier is improved through this study. We applied the operations of approximations to represent the imprecise knowledge accurately, and the concept of approximation quality is first applied in this method. Thus, this paper provides a novel rough set theory based prior probability in classical Bayesian classifier and the corresponding rough set prior Bayesian classifier. And we chose 18 public datasets to evaluate the performance of the proposed model compared with the classical Bayesian classifier and Bayesian classifier with Dirichlet prior assumption. Sufficient experimental results verified the effectiveness of the proposed method. The mainly impacts of our proposed method are: firstly, it provides a novel methodology which combines the rough set theory with the classical probability theory; secondly, it improves the accuracy of prior assumptions; thirdly, it provides an appropriate prior probability to the classical Bayesian classifier which can improve its performance only by improving the accuracy of prior assumption and without any effect to the likelihood probability; fourthly, the proposed method provides a novel and effective method to deal with the imprecise and uncertain data; last but not least, this methodology can be extended and applied to other concepts of classical probability theory, which providing a novel methodology to the probability theory.

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