scholarly journals Probability Integral as a Linearization

2018 ◽  
Vol 72 (1) ◽  
pp. 1-15
Author(s):  
Dušana Babicová
Keyword(s):  
Probability Space ◽  
Initial Structure ◽  
Unique Extension ◽  
Classical Probability ◽  
Sequential Convergence ◽  
Epireflective Subcategory ◽  
Random Events ◽  
Continuous Maps ◽  
Probability Integral ◽  
Fuzzy Random

Abstract In fuzzified probability theory, a classical probability space (Ω, A, p) is replaced by a generalized probability space (Ω, ℳ(A), ∫(.) dp), where ℳ(A) is the set of all measurable functions into [0,1] and ∫(.)dp is the probability integral with respect to p. Our paper is devoted to the transition from p to ∫(.) dp. The transition is supported by the following categorical argument: there is a minimal category and its epireflective subcategory such that A and ℳ(A) are objects, probability measures and probability integrals are morphisms, ℳ(A) is the epireflection of A, ∫(.) dp is the corresponding unique extension of p, and ℳ(A) carries the initial structure with respect to probability integrals. We discuss reasons why the fuzzy random events are modeled by ℳ(A) equipped with pointwise partial order, pointwise Łukasiewicz operations (logic) and pointwise sequential convergence. Each probability measure induces on classical random events an additive linear preorder which helps making decisions. We show that probability integrals can be characterized as the additive linearizations on fuzzy random events, i.e., sequentially continuous maps, preserving order, top and bottom elements.

Divisible extension of probability

2020 ◽  
Vol 70 (6) ◽  
pp. 1445-1456
Author(s):  
Roman Frič ◽  
Peter Eliaš ◽  
Martin Papčo
Keyword(s):  
Category Theory ◽  
Probability Space ◽  
Boolean Logic ◽  
Conditional Probabilities ◽  
Real Numbers ◽  
Classical Probability ◽  
Random Event ◽  
Whole Numbers ◽  
Random Events ◽  
Probability Integral

AbstractWe outline the transition from classical probability space (Ω, A, p) to its "divisible" extension, where (as proposed by L. A. Zadeh) the σ-field A of Boolean random events is extended to the class 𝓜(A) of all measurable functions into [0,1] and the σ-additive probability measure p on A is extended to the probability integral ∫(·) dp on 𝓜(A). The resulting extension of (Ω, A,p) can be described as an epireflection reflecting A to 𝓜(A) and p to ∫(·) dp.The transition from A to 𝓜(A), resembling the transition from whole numbers to real numbers, is characterized by the extension of two-valued Boolean logic on A to multivalued Łukasiewicz logic on 𝓜(A) and the divisibility of random events: for each random event u ∈ 𝓜(A) and each positive natural number n we have u/n ∈ 𝓜(A) and ∫(u/n) dp = (1/n) ∫u dp.From the viewpoint of category theory, objects are of the form 𝓜(A), morphisms are observables from one object into another one and serve as channels through which stochastic information is conveyed.We study joint random experiments and asymmetrical stochastic dependence/independence of one constituent experiment on the other one. We present a canonical construction of conditional probability so that observables can be viewed as conditional probabilities.In the present paper we utilize various published results related to "quantum and fuzzy" generalizations of the classical theory, but our ultimate goal is to stress mathematical (categorical) aspects of the transition from classical to what we call divisible probability.

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.

scholarly journals Study of T-norms and Quantum Logic Functions on BL-algebra and Their Relationships to the Classical Probability Structures

2020 ◽  
pp. 591-599
Author(s):  
Ahmed AL-Adilee ◽  
Habeeb Kareem Abdullah ◽  
Hawraa A. AL-Challabi
Keyword(s):  
Quantum Logic ◽  
Probability Space ◽  
Binary Operation ◽  
The Other ◽  
Symmetric Difference ◽  
Logic Function ◽  
Classical Probability ◽  
Binary Operations ◽  
Basic Probability ◽  
Logic Functions

This paper is concerned with the study of the T-norms and the quantum logic functions on BL-algebra, respectively, along with their association with the classical probability space. The proposed constructions depend on demonstrating each type of the T-norms with respect to the basic probability of binary operation. On the other hand, we showed each quantum logic function with respect to some binary operations in probability space, such as intersection, union, and symmetric difference. Finally, we demonstrated the main results that explain the relationships among the T-norms and quantum logic functions. In order to show those relations and their related properties, different examples were built.

scholarly journals Real Functions and the Extension of Generalized Probability Measures

2013 ◽  
Vol 55 (1) ◽  
pp. 85-94
Author(s):  
Jana Havlíčková
Keyword(s):  
Category Theory ◽  
Continuous Functions ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Classical Probability ◽  
Random Events ◽  
Elementary Methods ◽  
Generalized Probability ◽  
Fuzzy Probability Theory

Abstract In the classical probability, as well as in the fuzzy probability theory, random events and probability measures are modelled by functions into the closed unit interval [0,1]. Using elementary methods of category theory, we present a classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures) from a suitable class of generalized random events to a larger class having some additional (algebraic and/or topological) properties. The classification puts into a perspective the classical and some recent constructions related to the extension of sequentially continuous functions.

Measurements and Consistent Families of Quantum Channels

2011 ◽  
Vol 18 (03) ◽  
pp. 235-251
Author(s):  
Yves Le Jan ◽  
Rolando Rebolledo
Keyword(s):  
Quantum System ◽  
Moment Problem ◽  
Probability Space ◽  
Measurement Instrument ◽  
Quantum Systems ◽  
Quantum Channels ◽  
Simultaneous Observation ◽  
Classical Probability ◽  
Quantum Version ◽  
The Moment

This article introduces the notion of consistent families (Λ(n))n≥1of quantum channels. These families correspond to simultaneous observation of different copies of a given quantum system. Here, we are primarily interested in the analysis of measurements connected with them. As usual, the measurement of a quantum system requires the construction of a classical dilation of the corresponding quantum channel. In our case, the quantum systems represented by (Λ(n))n≥1are supposed to interact through the measurement instrument only. That is, we construct a classical probability space which allows to have a common dilation for all the Λ(n)' s . Doing this, we introduce and solve a quantum version of the moment problem.

PERCEPTION-BASED ESTIMATIONS OF FUZZY RANDOM VARIABLES: LINEARITY AND CONVEXITY

2008 ◽  
Vol 16 (supp01) ◽  
pp. 71-87 ◽  
Author(s):  
YUJI YOSHIDA
Keyword(s):  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Fuzzy Random Variables ◽  
Engineering Economics ◽  
Random Events ◽  
Fuzzy Random

A set of perceived random events is given by a fuzzy random variable, and an estimation of real random variables is represented by a functional on real random variables. The perception-based extension of estimation regarding random events is introduced, extending the functional to a functional of fuzzy random variables. This paper discusses some conditions and various properties of the extended estimations, for example, monotonicity, continuity, linearity, sub-additivity/super-additivity, convexity/concavity. Several examples of the perception-based extended estimations are investigated. This paper analyzes the general cases, where the estimations do not have monotone properties, from the viewpoint of convexity/concavity. The results can be applicable to other estimations in engineering, economics and so on.

scholarly journals Copulas in Classical Probability Sense

2020 ◽  
Vol 17 (3) ◽  
pp. 0889
Author(s):  
Ahmed AL-Adilee ◽  
Zainalabideen Samad ◽  
Samer Al-Shibley
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Joint Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Distribution Functions ◽  
Classical Probability ◽  
Probability Relation ◽  
Equivalent Probability Measure

               Copulas are simply equivalent structures to joint distribution functions. Then, we propose modified structures that depend on classical probability space and concepts with respect to copulas. Copulas have been presented in equivalent probability measure forms to the classical forms in order to examine any possible modern probabilistic relations. A probability of events was demonstrated as elements of copulas instead of random variables with a knowledge that each probability of an event belongs to [0,1]. Also, some probabilistic constructions have been shown within independent, and conditional probability concepts. A Bay's probability relation and its properties were discussed with respect to copulas. Moreover, an extension of multivariate constructions of each probabilistic copula has been presented. Finally, we have shown some examples that explain each relation of copula in terms of probability space instead of distribution functions.

Using the Concept of Fuzzy Random Events in the Assessment and Analysis of Ecological Risks

2015 ◽  
Vol 3 ◽  
pp. 193
Author(s):  
Oleg Uzhga-Rebrov ◽  
Galina Kuleshova
Keyword(s):  
Statistical Data ◽  
External Environment ◽  
Ecological Risks ◽  
Sources Of Information ◽  
Specific System ◽  
Initial Information ◽  
Subjective Evaluations ◽  
Random Events ◽  
Complicated Task ◽  
Fuzzy Random

<p>In many cases, the assessment<em> </em>and analysis of ecological risks is a complicated task, which is first of all related to obtaining reliable initial information. As a rule, ecological risks are due to unrepeated unique situations; from this it follows that sufficient statistical data on whose basis reliable evaluation of specific risks is made, are not available. On the other hand, unfavourable impacts on the external environment can affect the components of an ecosystem differently. The complexity of correlations among the components of an ecosystem significantly complicates an analysis of possible impacts on the components of a specific system.</p><p>When statistical data are missing or insufficient, experts who perform the required assessment on the basis of their knowledge and experience but often also using their intuition, are the only source of initial data. Here, however, the problem of reliability of expert evaluations arises. If other sources of information are missing, we have to accept subjective evaluations of experts as a basis, without an opportunity to evaluate the degree of their confidence.</p><p class="R-AbstractKeywords">In this kind of situation, it seems to be validated to introduce the extent of uncertainty into the evaluations of parameters of ecological risks. This can be accomplished by using fuzzy initial evaluations. This paper focuses on the concept of fuzzy random events and shows favourable chances of using that concept in the assessment and analysis of ecological risks. </p>

scholarly journals Real functions and the extension of generalized probability measures II

2015 ◽  
Vol 62 (1) ◽  
pp. 191-204
Author(s):  
Jana Havlíčková
Keyword(s):  
Probability Theory ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Random Events ◽  
Real Functions ◽  
Generalized Probability ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random

Abstract We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

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