Kolmogorov Probability Research Articles

Stability of a multi-step recovery process when catastrophe varying intensity

The actual problem of simulating the operation of the "Man-Machine-Environment" system, that is the process of the object recovery after the environmental disaster is considered, provided the recovery is made by one of its sub-systems, that include humans. The model differs significantly from the classical theory of reliability. The work is devoted to modeling of multi-step restoration process of an arbitrary nature object with non-stationary Poisson stream of events (accidents) and exponential intensity of recovery process. It passes a fixed finite sequence of phases - the states and is described by the Kolmogorov probabilities for these states. The cases of ergodic and absorbing chains with continuous time are considered. Some of the states in the chain indicate the efficiency of the operator in elimination the accident. It is assumed that the efficiency of the operator can not recover during the process of eliminating accidents. According to the verbal descriptions of the object, graphs of states are drawn, and in accordance to them – the Kolmogorov equations and their stationary solutions. The resulting figures of numerical solutions allow us to determine the time of the process stabilization. For actual input data the following resulting probabilities are obtained: for trouble-free operation of the facility, for a fatal accident and for disaster recovery.

Realization of the probability laws in the quantum central limit theorems by a quantum walk

Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.

The digital randomness of black-body radiation

The statistical properties of the fractional part of the random energy of a spectral component of black-body radiation have been analysed in the frame of classical Kolmogorovian probability theory. Besides the integer part of the energy (which satisfies the well-known Planck- Bose distribution) the realizations of its fractional part (related to 'round-off errors') has been represented by binary sequences, like z = 0.001011000010.... It has been shown that the binary variables realized by the 0-s and 1-s at different positions are independent. From the condition of independence the original distribution of the fractional part z can be recovered. If these binary variables have the same distribution, they describe a temperature-independent (random) energy, whose expectation value is the well-known zero-point energy. Thus, the zero-point fluctuations can be considered as a physical representative of an ideal random number generator.

Markov model of unsteady flow elimination of accidents in restrictions on the performance of the operator

The actual problem of simulating the operation of the Man-Machine-Environment system, that is the process of the object recovery after the environmental disaster is considered, provided the recovery is made by one of its sub-systems, that include humans. The model differs significantly from the classical theory of reliability. The work is devoted to modeling of multi-step restoration process of an arbitrary nature object with non-stationary Poisson stream of events (accidents) and exponential intensity of recovery process. It passes a fixed finite sequence of phases - the states and is described by the Kolmogorov probabilities for these states. The cases of ergodic and absorbing chains with continuous time are considered. Some of the states in the chain indicate the efficiency of the operator in elimination the accident. It is assumed that the efficiency of the operator can not recover during the process of eliminating accidents. According to the verbal descriptions of the object, graphs of states are drawn, and in accordance to them – the Kolmogorov equations and their stationary solutions. The resulting figures of numerical solutions allow us to determine the time of the process stabilization. For actual input data the following resulting probabilities are obtained: for trouble-free operation of the facility, for a fatal accident and for disaster recovery

Document ranking with quantum probabilities

In this thesis we investigate the use of quantum probability theory for ranking documents. Quantum probability theory is used to estimate the probability of relevance of a document given a user's query. We posit that quantum probability theory can lead to a better estimation of the probability of a document being relevant to a user's query than the common IR approach, i. e. the Probability Ranking Principle (PRP), which is based upon Kolmogorovian probability theory. Following our hypothesis, we formulate an analogy between the document retrieval scenario and a physical scenario, that of the double slit experiment. Through the analogy, we propose a novel ranking approach, the quantum probability ranking principle (qPRP). Key to our proposal is the presence of quantum interference. Mathematically, this is the statistical deviation between empirical observations and expected values predicted by the Kolmogorovian rule of additivity of probabilities of disjoint events in configurations such that of the double slit experiment. While PRP explicitly assumes that the relevancy of a document is independent of that of other documents, we suggest that qPRP implicitly models interdependent document relevance through quantum interference and thus is suited to those document ranking tasks where the independence assumption fails. Throughout the thesis, we also suggest how quantum interference can be estimated for effective document ranking. To validate our proposal and to gain more insights about approaches for document ranking, we (1) analyse PRP, qPRP and other ranking approaches, exposing the assumptions underlying their ranking criteria and formulating the conditions for the optimality of the two ranking principles, (2) empirically compare three ranking principles (i. e. PRP, interactive PRP, and qPRP) and two state-of-the-art ranking strategies in two retrieval scenarios, those of ad-hoc retrieval and diversity retrieval, (3) analytically contrast the ranking criteria of the examined approaches, exposing similarities and differences, (4) study the ranking behaviours of approaches alternative to PRP in terms of the kinematics they impose on relevant documents, i. e. by considering the extent and direction of the movements of relevant documents across the ranking recorded when comparing PRP against its alternatives. Our findings show that the effectiveness of the examined ranking approaches strongly depends upon the evaluation context. In the traditional evaluation context of ad-hoc retrieval, PRP is empirically shown to be better than or comparable to alternative ranking approaches. However, when evaluation contexts that account for interdependent document relevance are examined (i. e. when the relevance of a document is assessed also with respect to other retrieved documents, as it is the case in the diversity retrieval scenario), the use of quantum probability theory and thus of qPRP is shown to improve retrieval and ranking effectiveness over the traditional PRP and alternative ranking strategies, such as Maximal Marginal Relevance, Portfolio theory, and Interactive PRP. This work represents a significant step forward regarding the use of quantum theory in information retrieval. It demonstrates that the application of quantum theory to problems within information retrieval can lead to improvements both in modelling power and retrieval effectiveness, allowing the constructions of models that capture the complexity of information retrieval situations. Furthermore, the thesis opens up a number of lines of future research. These include investigating estimations and approximations of quantum interference in qPRP, exploiting complex numbers for the representation of documents and queries, and applying the concepts underlying qPRP to tasks other than document ranking. This dissertation was completed at School of Computing Science, University of Glasgow under the advise of Dr. Leif Azzopardi and Prof. Keith van Rijsbergen. Prof. Norbert Fuhr, Dr. Iadh Ounis, and Dr. John O'Donnell served as dissertation committee members. For the full dissertation, visit: http://theses.gla.ac.uk/3463.

Quantum probabilities in competing lizard communities

AbstractDespite predictive success, population dynamics and evolutionary game theory still pose fundamental problems. Violation of the competitive exclusion principle in plankton communities provides an example. A promising solution of this ‘paradox of the plankton’ comes from theories involving cyclic competition, an evolutionary analogue of the classical rock-paper-scissors (RPS) game. However, modeling probabilistic RPS structures, one encounters a fundamental difficulty: the pairs rock-scissors, scissors-paper, and paper-rock possess representations in separate Kolmogorovian probability spaces, but a single global probability space for entire triplets does not exist. Populations that take part in cyclic competition should therefore involve probabilistic incompatibilities, analogous to those occurring in quantum mechanics. Here, using experimental data collected from 1990 to 2011 on the RPS cycles of lizards, we show that the incompatibilities are indeed unavoidable, and the data cannot be reconstructed from a single Kolmogorovian probability space. We then prove that the effect is genuinely quantum probabilistic, i.e. all the probabilities can be formulated in terms of a single density matrix and a set of non-commuting projectors. This formal quantum structure is dormant in games where probabilities of strategies do not entangle with probabilities of payoffs, and thus could be overlooked. In more realistic scenarios, involving games ‘with ace in a sleeve’, the non-Kolmogorovian structure can be activated. Surprisingly, lizards occasionally do play such games. In consequence, the formalism of evolutionary games, similarly to quantum mechanics, should begin with density matrix equations. Implications of our finding extend beyond lizard communities, given that RPS games are common in nature and higher dimensional RPS games may be even more common in ecosystems.

Contingent Hybrid Systems

This work proposes the first steps for studying resilient control for uncertain hybrid systems. We take a rather radical approach and consider the situations of severe uncertainty. This requires the replacement of classical Kolmogorov probabilities with special concepts from the generalized measure theory. To achieve this goal, we need a rigorous and carefully developed mathematical framework, which constitutes the major contribution of this paper. Technically, the resilience framework of hybrid systems is defined in two steps. First, we introduce the contingent hybrid system that is a mathematical model where uncertainty is quantized using both probabilities and generalized measures. Then, the concept of lithe decision is introduced for this model. The new method combines decision theory with model predictive control and statistics for uncertainty measures.

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables

Some Kolmogorov probability inequalities for quadratic forms and weighted quadratic forms of negative superadditive dependent (NSD) uniformly bounded random variables are provided. Using these inequalities, some complete convergence of randomized quadratic forms under some suitable conditions are evaluated. Moreover, various examples are presented in which the given conditions of our results are satisfied.

When Quantum Mechanics Interacts with Cognitive Science

I reflect on several aspects of the general claim that a quantum-like approach to Cognitive Science is advantageous over classical approaches. The classical approaches refer to the symbolic approaches including models using a classical (Kolmogorov) probability calculus. The general claim seems to be right from a descriptive viewpoint but not necessarily from an explanatory viewpoint. The explanatory perspective needs a more careful analysis since adding some additional arbitrary parameters (such as phase shift parameters in quantum probabilities) does not automatically increase the explanatory value of the approach; rather, it seems to decrease it. I argue further that there is another class of traditional models – the class of geometric models of cognition. These models have a much longer tradition than the symbolic models. Interestingly, quantum mechanics does not contradict the geometric models. Hence, real progress at the meeting between quantum mechanics and cognitive science could be made by unifying these geometric models with ideas from quantum theory.

Quantum Mechanics Meets Cognitive Science: Explanatory vs Descriptive Approaches

We reflect on several aspects of the general claim that a quantum‐like approach to Cognitive Science is advantageous over classical approaches. The classical approaches refer to the symbolic approaches including models using a classical (Kolmogorov) probability calculus. The general claim seems to be right from a descriptive viewpoint but not necessarily from an explanatory viewpoint. The explanatory perspective needs a more careful analysis since adding some additional arbitrary parameters (such as phase shift parameters in quantum probabilities) does not automatically increase the explanatory value of the approach; rather, it seems to decrease it. We argue further that there is another class of traditional models – the class of geometric models of cognition. These models have a much longer tradition than the symbolic models. Interestingly, quantum mechanics does not contradict the geometric models. Hence, real progress at the meeting between quantum mechanics and cognitive science could be made by unifying these geometric models with ideas from quantum theory.

Quantum Covers in Quantum Measure Theory

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, the Kolmogorov sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers.

Individual events and mathematical formalism of quantum mechanics

A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.

Epr-bohm experiment and Bell’s inequality: Quantum physics meets probability theory

Our main aim in this paper is to inform the physics community (and especially experts in quantum information) about investigations of the problem of the probabilistic compatibility of a family of random variables: the possibility of realizing such a family based on a single probability measure (of constructing a single Kolmogorov probability space). These investigations were started a hundred years ago by Boole. The complete solution of the problem was obtained by the Soviet mathematician Vorobiev in the 1960s. It turns out that probabilists and statisticians obtained inequalities for probabilities and correlations that include the famous Bell’s inequality and its generalizations.

Nonlocality as well as rejection of realism are only sufficient (but non-necessary!) conditions for violation of Bell’s inequality

In this review we remind the viewpoint that violation of Bell’s inequality might be interpreted not only as an evidence of the alternative – either nonlocality or “death of reality” (under the assumption the quantum mechanics is incomplete). Violation of Bell’s type inequalities is a well known sufficient condition of probabilistic incompatibility of random variables – impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell’s inequality – a few pairs of random variables (two-dimensional vector variables) involved in the EPR–Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus, one can choose between: (a) completeness of quantum mechanics; (b) nonlocality; (c) “ death of reality”; (d) non-Kolmogorovness. In any event, violation of Bell’s inequality has a variety of possible interpretations. Hence, it could not be used to obtain the definite conclusion on the relation between quantum and classical models.

A Kolmogorov inequality for weighted U-statistics

In this paper a Kolmogorov probability inequality for weighted U-statistics based on Bernoulli kernels is presented. This inequality which extends the results of [Turner, D.W., Young, D.M., Seaman, J.W., 1995. A Kolmogorov inequality for the sum of independent Bernoulli random variables with unequal means. Statist. Probab. Lett. 23, 243–245] is a Hoeffding type exponential inequality without any assumptions or restrictions.

A DECISION PROCEDURE FOR PROBABILITY CALCULUS WITH APPLICATIONS

A decision procedure (PrSAT) for classical (Kolmogorov) probability calculus is presented. This decision procedure is based on an existing decision procedure for the theory of real closed fields, which has recently been implemented inMathematica. AMathematicaimplementation of PrSAT is also described, along with several applications to various non-trivial problems in the probability calculus.

Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?

The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell’s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and “death of reality” which are typically linked to Bell’s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell’s type inequalities statistical data from a number of experiments performed under different experimental contexts.

How To Play Two-Player Restricted Quantum Games with 10 Cards

We show that it is possible to play ‘restricted’ two-player quantum games proposed originally by Marinatto and Weber (Phys. Lett. A 272:291–303, 2000) by purely macroscopic means, in the simplest case having as the only equipment a pack of 10 cards. Our example shows also that some apparently ‘genuine quantum’ results, even those that emerge as a consequence of dealing with entangled states, can be obtained by suitable application of Kolmogorovian probability calculus and secondary-school mathematics, without application of the ‘Hilbert space machinery’.

Measurements and mathematical formalism of quantum mechanics

A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.

A Formula of Total Probability with the Interference Term and the Hilbert Space Representation of the Contextual Kolmogorovian Model

We compare the classical Kolmogorov and quantum probability models. We show that the gap between these models is not so huge as was commonly believed. The main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the Kolmogorov model. In particular, we obtain “interference of probabilities” without appealing to the Hilbert space formalism. We interpret “interference of probabilities” as a perturbation (by a cos-term) of the conventional formula of total probability. Our classical derivation of quantum probabilistic formalism can stimulate applications of quantum methods outside of the microworld, for instance, in psychology, biology, economy, and other domains of science.