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 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.

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.

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 Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 199 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle Jorgensen
Keyword(s):  
Probability Space ◽  
Free Probability ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Probabilistic Information ◽  
Closed Intervals ◽  
Measurable Functions ◽  
Semicircular Elements

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .

Dempster Combination Rule for Signed Belief Functions

1998 ◽  
Vol 06 (01) ◽  
pp. 79-102 ◽  
Author(s):  
Ivan Kramosil
Keyword(s):  
Large Class ◽  
Probability Space ◽  
Binary Operation ◽  
Random Variables ◽  
Belief Function ◽  
Measurable Space ◽  
Point Of View ◽  
Belief Functions ◽  
Combination Rule ◽  
Real Numbers

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

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.

On the Equivalence of Modes of Convergence(1)

1973 ◽  
Vol 16 (4) ◽  
pp. 571-575 ◽  
Author(s):  
R. J. Tomkins
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Real Numbers ◽  
Probability Zero

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

Multiple analogical proportions

2021 ◽  
pp. 1-18
Author(s):  
Henri Prade ◽  
Gilles Richard
Keyword(s):  
Artificial Intelligence ◽  
Linear Algebra ◽  
Building Block ◽  
Analogical Reasoning ◽  
Weak Form ◽  
Boolean Logic ◽  
Real Numbers ◽  
Analogical Inference ◽  
Logical Modeling

Analogical proportions are statements of the form “a is to b as c is to d”, denoted a : b : : c : d, that may apply to any type of items a, b, c, d. Analogical proportions, as a building block for analogical reasoning, is then a tool of interest in artificial intelligence. Viewed as a relation between pairs ( a , b ) and ( c , d ), these proportions are supposed to obey three postulates: reflexivity, symmetry, and central permutation (i.e., b and c can be exchanged). The logical modeling of analogical proportions expresses that a and b differ in the same way as c and d, when the four items are represented by vectors encoding Boolean properties. When items are real numbers, numerical proportions – arithmetic and geometric proportions – can be considered as prototypical examples of analogical proportions. Taking inspiration of an old practice where numerical proportions were handled in a vectorial way and where sequences of numerical proportions of the form x 1 : x 2 : ⋯ : x n : : y 1 : y 2 : ⋯ : y n were in use, we emphasize a vectorial treatment of Boolean analogical proportions and we propose a Boolean logic counterpart to such sequences. This provides a linear algebra calculus of analogical inference and acknowledges the fact that analogical proportions should not be considered in isolation. Moreover, this also leads us to reconsider the postulates underlying analogical proportions (since central permutation makes no sense when n ⩾ 3) and then to formalize a weak form of analogical proportion which no longer obeys the central permutation postulate inherited from numerical proportions. But these weak proportions may still be combined in multiple weak analogical proportions.

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.

Diffy

1971 ◽  
Vol 18 (6) ◽  
pp. 402-406
Author(s):  
Herbert Wills
Keyword(s):  
Individual Differences ◽  
Grade Level ◽  
Real Numbers ◽  
Whole Numbers

The game DIFFY is an academic game that provides intrinsically interesting drill experiences and allows for individual differences. Moreover, teachers need not construct any of the exercises. The student himself initiates the drill, and the nature of the game provides for variety in the numbers encountered. Thus the activity promotes self-generated drill. This feature is commendable, since students won't run out of material and they become personally involved in their work. Besides automatically adjusting to students of diverse abilities within a given classroom, DIFFY also provides productive academic recreation at every grade level. This flexibility stems from the fact that the game may be played with a variety of numbers: whole numbers, nonnegative rationals, integers, all rationals, or all real numbers.

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