On probabilistic argumentation and subargument-completeness

2021 ◽  
Author(s):  
Régis Riveret ◽  
Nir Oren
Keyword(s):  
Probability Theory ◽  
Probability Space ◽  
Short Communication ◽  
Sample Space ◽  
Formal Models ◽  
Probabilistic Argumentation

Abstract Probabilistic argumentation combines probability theory and formal models of argumentation. Given an argumentation graph where vertices are arguments and edges are attacks or supports between arguments, the approach of probabilistic labellings relies on a probability space where the sample space is any specific set of argument labellings of the graph, so that any labelling outcome can be associated with a probability value. Argument labellings can feature a label indicating that an argument is not expressed, and in previous work these labellings were constructed by exploiting the subargument-completeness postulate according to which if an argument is expressed then its subarguments are expressed and through the use of the concept of ‘subargument-complete subgraphs’. While the use of such subgraphs is interesting to compare probabilistic labellings with other works in the literature, it may also hinder the comprehension of a relatively simple framework. In this short communication, we revisit the construction of probabilistic labellings and demonstrate how labellings can be specified without reference to the concept of subargument-complete subgraphs. By doing so, the framework is simplified and yields a more natural model of argumentation.

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Diagrams for difficult problems in probability

2003 ◽  
Vol 87 (508) ◽  
pp. 86-97 ◽  
Author(s):  
Peter C.-H. Cheng ◽  
Nigel G. Pitt
Keyword(s):  
Probability Theory ◽  
Conceptual Understanding ◽  
Probability Space ◽  
Learning And Instruction

We have developed a novel diagrammatic approach for understanding and teaching probability theory — Probability Space diagrams [1]. Our studies of learning and instruction with Probability Space (PS) diagrams have demonstrated that they can significantly enhance students' conceptual understanding. This article illustrates the utility of PS diagrams by applying them to the explanation of some difficult concepts and notoriously counterintuitive problems in probability. We first outline the nature of the system.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

The two-dimensional Poisson process and extremal processes

1971 ◽  
Vol 8 (4) ◽  
pp. 745-756 ◽  
Author(s):  
James Pickands
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Order Statistics ◽  
Sample Space ◽  
Two Dimensional ◽  
Record Times ◽  
Extreme Order Statistics ◽  
Extremal Processes

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

Fuzzy Quantum Space

2017 ◽  
pp. 84-114
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Boolean Algebra ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Space ◽  
Quantum Space ◽  
Fuzzy State ◽  
Fuzzy Observable

In this chapter we study the existence of a sum of fuzzy observables in a fuzzy quantum space which generalizes the Kolmogorov probability space using the ideas of fuzzy set theory. We also study some properties of the sum of fuzzy observables. To study the above mentioned, we also include the basic notions from the probability theory on fuzzy quantum space in this chapter, i.e. the notion of fuzzy quantum space, a fuzzy observable, an indicator of a fuzzy set, a null fuzzy observable, a Boolean algebra on fuzzy quantum space, fuzzy state etc.

Explainable Acceptance in Probabilistic Abstract Argumentation: Complexity and Approximation

2020 ◽  
Author(s):  
Gianvincenzo Alfano ◽  
Marco Calautti ◽  
Sergio Greco ◽  
Francesco Parisi ◽  
Irina Trubitsyna
Keyword(s):  
Probability Theory ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Approximation ◽  
Special Class ◽  
Hard Problem ◽  
Argumentation Framework ◽  
Abstract Argumentation ◽  
Additive Error ◽  
Probabilistic Argumentation

Recently there has been an increasing interest in probabilistic abstract argumentation, an extension of Dung's abstract argumentation framework with probability theory. In this setting, we address the problem of computing the probability that a given argument is accepted. This is carried out by introducing the concept of probabilistic explanation for a given (probabilistic) extension. We show that the complexity of the problem is FP^#P-hard and propose polynomial approximation algorithms with bounded additive error for probabilistic argumentation frameworks where odd-length cycles are forbidden. This is quite surprising since, as we show, such kind of approximation algorithm does not exist for the related FP^#P-hard problem of computing the probability of the credulous acceptance of an argument, even for the special class of argumentation frameworks considered in the paper.

The two-dimensional Poisson process and extremal processes

1971 ◽  
Vol 8 (04) ◽  
pp. 745-756 ◽  
Author(s):  
James Pickands III
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Order Statistics ◽  
Sample Space ◽  
Two Dimensional ◽  
Record Times ◽  
Extreme Order Statistics ◽  
Extremal Processes

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

scholarly journals A probability space of continuous-time discrete value stochastic process with Markov property

2016 ◽  
Vol 12 (3) ◽  
pp. 5975-5991
Author(s):  
Miloslawa Sokol
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Probability Space ◽  
Measure Theory ◽  
Markov Property ◽  
Finite Measure ◽  
Physical Models

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

On Multiple Integral Geometric Integrals and Their Applications to Probability Theory

1970 ◽  
Vol 22 (1) ◽  
pp. 151-163 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Integral Geometry ◽  
Mathematical Theory ◽  
Probability Space ◽  
Convex Bodies ◽  
Multiple Integral ◽  
Mean Value ◽  
Dimensional Euclidean Space ◽  
Point Sets

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.

scholarly journals Probability Theory and Its Models

2019 ◽  
pp. 168-181
Author(s):  
Paul Humphreys
Keyword(s):  
Probability Theory ◽  
Mathematical Models ◽  
Mathematical Theory ◽  
Scientific Theory ◽  
Formal Theory ◽  
Empirical Content ◽  
Formal Models ◽  
The Status ◽  
Historical Transition ◽  
Abstract Probability

This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. Some remarks are made about the historical transition from Hilbert’s view of probability as a scientific theory to Kolmogorov’s view of probability as a mathematical theory. A process is provided that bridges abstract probability theory with concrete systems via mathematical models. This demonstrates that empirical content is injected into formal models via the mapping from those formal models on to elements of the concrete systems. David Freedman and Philip Stark’s concept of model-based probabilities is examined and is used as a bridge between the formal theory and applications.

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