Probability Theory Research Articles

Bidirectional Feedback Mechanism in Group Decision-Making: A Quantum Probability Theory Model Based on Interference Effects

Bidirectional Feedback Mechanism in Group Decision-Making: A Quantum Probability Theory Model Based on Interference Effects

Probability via Expectation Measures

Since the seminal work of Kolmogorov, probability theory has been based on measure theory, where the central components are so-called probability measures, defined as measures with total mass equal to 1. In Kolmogorov’s theory, a probability measure is used to model an experiment with a single outcome that will belong to exactly one out of several disjoint sets. In this paper, we present a different basic model where an experiment results in a multiset, i.e., for each of the disjoint sets we obtain the number of observations in the set. This new framework is consistent with Kolmogorov’s theory, but the theory focuses on expected values rather than probabilities. We present examples from testing goodness-of-fit, Bayesian statistics, and quantum theory, where the shifted focus gives new insight or better performance. We also provide several new theorems that address some problems related to the change in focus.

Existence of Long-Range Order in Random-Field Ising Model on Dyson Hierarchical Lattice

We study the random-field Ising model on a Dyson hierarchical lattice, where the interactions decay in a power-law-like form, J(r)∼r-α, with respect to the distance. Without a random field, the Ising model on the Dyson hierarchical lattice has a long-range order at finite low temperatures when 1<α<2. In this study, for 1<α<3/2, we rigorously prove that there is a long-range order in the random-field Ising model on the Dyson hierarchical lattice at finite low temperatures, including zero temperature, when the strength of the random field is sufficiently small but nonzero. Our proof is based on Dyson’s method for the case without a random field, and the concentration inequalities in probability theory enable us to evaluate the effect of a random field.

Probabilistic theory of the L-H transition and causality

Probabilistic theory of the L-H transition and causality

ОПРЕДЕЛЕНИЕ УРОВНЯ ЭКСПЛУАТАЦИОННОЙ НАДЕЖНОСТИ ТРАНСПОРТНЫХ СРЕДСТВ

The study objective is to determine the level of operational reliability of the company's vehicles. The problem to which the paper is devoted is to develop an algorithm for calculating the influence of factors on the system of operational reliability of vehicles. Research methods are forecasting, experiment, mathematical modeling, statistical analysis, system analysis, machine aging theory, reliability theory, probability theory, control theory. The novelty of the work. A technique is developed to control operational reliability, which includes an algorithm for calculating the influence of factors on the operational reliability system. Study results. One of the main scientific tasks is to develop a technique for forecasting and assessing the level of operational reliability of vehicles as an indicator characterizing the ability of an enterprise to ensure the readiness of vehicles to fulfill their tasks. Conclusions: Implementation of the developed algorithm in motor transport enterprises makes it possible to determine the level of operational reliability of vehicles, including the provision of restoring the level of operational reliability to its maximum value.

Bayes Theorem in Machine Learning: A Literature Review

In the era of big data and articial intelligence, machine learning has become a crucial tool for extracting insights and making predictions across various domains. Bayes theorem, a fundamental principle in probability theory, has emerged as a cornerstone in many machine learning algorithms. This literature review explores the main applications of Bayes theorem in machine learning, focusing on its role in classication, Natural Language Processing (NLP), and other emerging elds. The study aims to provide an overview of how Bayesian principles enhance learning algorithms, improve decision-making processes, and address complex problems in articial intelligence. Through a systematic review of academic papers from Google Scholar, this research synthesizes current knowledge on Bayesian methods in machine learning. The methodology involves dening the research scope, conducting a literature search using specic keywords, screening relevant studies, and analyzing the collected data. By examining diverse applications ranging from disease prediction to sentiment analysis, this review highlights the versatility and signicance of Bayes theorem in advancing machine learning techniques and their real-world implementations.

Mathematical Principles in Texas Hold'em Poker

Texas Hold'em is a skill and strategy game that combines elements of probability, combinatorics, and game theory. This paper explores the mathematical principles behind the game, with a focus on applying probability theory to decision-making and strategy development. Analyzing the win rates of various hand rankings, the expected value of different playstyles, and the impact of player behavior on game outcomes. By applying combinatorial analysis, we calculate the probability of hand strength and assess the strategic significance of bluffing and betting patterns. Additionally, we study game theory concepts such as Nash equilibrium and apply them to poker strategy. The most significant section is game theory. The insights gained from this mathematical approach provide a comprehensive understanding of optimal play and how mathematical rigor enhances decision-making in Texas Hold'em. This research can help us calculate the probability of our hand based on its strength, thereby improving our sensitivity to probability-based problems

Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories

In a Generalised Probabilistic Theory (GPT) equipped additionally with some extra geometric structure we define the morphophoric measurements as those for which the measurement map sending states to distributions of the measurement results is a similarity. In the quantum case, morphophoric measurements generalise the notion of a 2-design POVM, thus in particular that of a SIC-POVM. We show that the theory built on this class of measurements retains the chief features of the QBism approach to the basis of quantum mechanics. In particular, we demonstrate how to extend the primal equation (&amp;apos;Urgleichung&amp;apos;) of QBism, designed for SIC-POVMs, to the morphophoric case of GPTs. In the latter setting, the equation takes a different, albeit more symmetric, form, but all the quantities that appear in it can be interpreted in probabilistic and operational terms, as in the original &amp;apos;Urgleichung&amp;apos;.

Connections between Statistics and Mathematics/Probability

There are many connections between probability, other mathematics courses, and statistics. Understanding these connections provides insights that might not be fully appreciated when considering each discipline in isolation. While the typical instruction of statistics courses relies on elucidating its foundational principles from mathematical and probability theory, it is generally less appreciated that statistics can in turn provide a deeper understanding of results in mathematics and probability. We offer several examples for which knowledge of statistics can shed new light on probability and other mathematics results. Examples span both undergraduate and graduate level material. In today’s data driven-world, many students are naturally curious about statistics and are exposed to this field early in their undergraduate curriculum. Leveraging connections between statistics and mathematics and probability makes theoretical concepts more intuitive and relevant, fostering a better understanding.

Probabilistic representation of the parabolic stochastic variational inequality with Dirichlet–Neumann boundary and variational generalized backward doubly stochastic differential equations

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with subdifferential of a lower semicontinuous convex function under a non-Lipschitz condition. This study allows us to give a probabilistic representation (in the stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet–Neumann conditions.

The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics.

We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.

Klasifikasi Penyakit Migrain dengan Metode Naïve Bayes pada Dataset Kaggle

Migraine is one of the most common neurological diseases in society and has a significant disability impact. According to the Global Burden of Disease Study, migraine is one of the most common neurological disorders in the world, with a greater disability burden than other neurological disorders. In data science, data classification plays an important role in determining the category or class of an object based on a number of available classes. One frequently used classification method is Naïve Bayes, which utilizes mathematical probabilities with the assumption that the decision made is accurate based on the given data. This research develops a classification model using the Naïve Bayes algorithm to predict and classify migraine patient data. This model produces classification values with an accuracy of 88.51% when training from 280 train data given and 89.02% when testing from 120 test data given. The classification results can support the medical world in diagnosing migraine types more accurately, optimizing treatment, saving health costs, and becoming the basis for further research and development in the field of neurology.

On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Their Fracterm Calculus.

Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols ±∞ for signed infinite values and a symbol named ⟂ for the undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback-Leibler divergence, and Shannon divergence can be expressed without concerning any further conventions. The algebra may form a basis for probability theory more generally.

Recurrence rates for shifts of finite type

Let ΣA be a topologically mixing shift of finite type, let σ:ΣA→ΣA be the usual left-shift, and let μ be the Gibbs measure for a Hölder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system (ΣA,σ) that hold μ-almost surely. In particular, given a function ψ:N→N we are interested in the following setRψ={i∈ΣA:in+1…in+ψ(n)+1=i1…iψ(n)for infinitely manyn∈N}.We provide sufficient conditions for μ(Rψ)=1 and sufficient conditions for μ(Rψ)=0. As a corollary of these results, we discover a new critical threshold where the measure of Rψ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets.

Accurate Predictions of Gravity without Mass, Null Dark Matter Results, Muon Precession and Solutions to the Hubble Tension, Final Parsec Problem, Information Paradox, Inter Alia: A 3.5-Year Status Report on the Probabilistic Spacetime Theory

Accurate Predictions of Gravity without Mass, Null Dark Matter Results, Muon Precession and Solutions to the Hubble Tension, Final Parsec Problem, Information Paradox, Inter Alia: A 3.5-Year Status Report on the Probabilistic Spacetime Theory

“Belief Updating under Uncertainty: Probabilistic Epistemology and Its Limitations”

Probabilistic epistemology, especially the Bayesian model, provides a dynamically updated framework for the construction of knowledge under uncertainty. Starting from the shortcomings of the traditional view of knowledge, this paper analyzes how probabilistic theory can cope with uncertainty by adjusting the strength of beliefs, so as to gradually approach the truth. Bayesian epistemology has shown its unique application potential in the fields of science, medicine and economic forecasting. Especially, Bayesian epistemology can enhance rational belief in dynamic environment through the update mechanism of conditional probability. Although Bayesian epistemology has many advantages, it has obvious limitations in its application to non-empirical fields such as ethics and metaphysics. This paper also explores the importance of Bayesianism in knowledge construction and compares it with other epistemological models such as evidentialism and dependability. This paper argues that although Bayesianism provides flexible instrumental support for knowledge construction, it still has great limitations in dealing with fuzzy and non-quantitative knowledge and ethical decision-making, which limits its application to broader epistemological problems. At the same time, it is pointed out that probabilistic epistemology can be combined with other methods to expand its application scope.

Research on Innovative Teaching Methods of Probability Theory and Mathematical Statistics under the Background of Internet

Probability theory and mathematical statistics is an important basic course of science and technology, economics and management, and has close contact with industry, agriculture, military and science and technology, so it is particularly important for students to study probability theory and mathematical statistics. The learning effect of students largely depends on the teaching methods of teachers. Under the background of the Internet, the emerging development of Internet education, in order to conform to the development of The Times and improve the quality of teaching, the innovative teaching methods of probability theory and mathematical statistics should also be studied. Based on the analysis of the characteristics and advantages of Internet education, combined with the characteristics of probability theory and mathematical statistics and the shortcomings of current teaching, this paper designs an innovative teaching method based on the application of flipped classroom teaching mode under the Internet. Taking the rain classroom as an example, the teaching method is designed from three links: preview before class, teaching in class and feedback after class. Under the background of Internet education, it is of great significance to improve the teaching quality of probability theory and mathematical statistics courses.

A Study of Post Keynesian Attempts at Hiding Townshend’s Main Question to Keynes in His November 1938 Letter and Keynes’s Answer

Keynes’s letter of December 7th, 1938, provided a direct answer to Townshend, who asked Keynes the following extremely important question in his letter of November 25th: “This is the nearest I can get to an analysis of the part played by the factor of confidence in the rationale of interest. I believe that its further logical analysis at a deeper level of generalization is connected with the part played by the weight of evidence in your theory of probability, but I cannot see just how…..” (Keynes 1979, 292; italics added). Now Townshend ‘s question actually is “Where in your A Treatise on Probability is your analysis supporting the connection between confidence in the GT and the weight of evidence?” Keynes’s response was direct and straightforward: “As regards my remarks in my General Theory, have you taken account of what I say on page 240, as well as what I say at page 148, which is the passage I think you previously quoted…”. (Keynes 1979; italics added). The clue, given here by Keynes to Townshend, is to p.240 of the General Theory; however, it relates directly, as we will see, to Keynes’s chapter XXVI of the A Treatise on Probability. This paper will trace out how Keynes provided Townshend with the clues needed to recognize Keynes’s modeling of the conventional coefficient in chapter XXVI; however, Townshend gave up and failed to take this last step.

ФУНКЦІЯ ЩІЛЬНОСТІ РОЗПОДІЛУ ЦІЛЕЙ ДЛЯ ПЛАНУВАННЯ ЗАСТОСУВАННЯ БЕЗПІЛОТНИХ ЛІТАЛЬНИХ АПАРАТІВ

The main problem of special aerial reconnaissance is the contradiction between the dimensions of the survey area and the dimensions of elementary objects (targets). Since the viewing area is very large, the size of the target is small, and the instantaneous field of view of the on-board equipment is limited, the time for obtaining data is quite significant and the amount of information to be processed is very large. Also, due to technical limitations on the time spent in the air and the limitations of the reconnaissance equipment, one UAV is not able to cover the entire area in one flight. The severity of this problem can be partially removed by using a group of UAVs. but both for one person and for a group, the task of planning the route of movement arises. A simple comb survey of the reconnaissance zone is far from optimal, and in practice route planning is used by points with the involvement of a priori information about the most likely locations of targets. However, this method is rather inconvenient for the automatic route planning of the UAV itself, since the point on the map usually does not correlate with the flight height, the instantaneous field of view of the on-board surveillance equipment, the scale and detail of the picture. In order to increase the optimization possibilities of automatic flight planning, it is suggested to use the target distribution density function (TDF). The density function of the target distribution is a two-dimensional mathematical model that describes the conditional relative probability of finding targets at different points in space. It is created and set on the basis of a priori data about the terrain, previous observations or intellectual assessments, reflecting the distribution of possible targets in a certain area or for the entire reconnaissance area. TDF allows to model the space not as homogeneous, but as an area with different degrees of importance or probabilities of finding targets. It should be noted that obtaining mathematically justified or calculated by the theory of probability or mathematical statistics, two-dimensional discrete probability densities for randomly placed targets is actually impossible. Therefore, here the TDF is considered as an area with different degrees of review importance, which is reflected by conditional relative probabilities obtained from various sources, including expert evaluation.

Realization of an arbitrary structure of perfect distinguishability of states in general probability theory

Let [Formula: see text] be states of a general probability theory, and [Formula: see text] be the set of all subsets of indices [Formula: see text] such that the states [Formula: see text] are jointly (perfectly) distinguishable. All subsets with a single element are of course in [Formula: see text], and since smaller collections are easier to distinguish, if [Formula: see text] and [Formula: see text] then [Formula: see text]; in other words, [Formula: see text] is a so-called independence system on the set of indices [Formula: see text]. In this paper, it is shown that every independence system on [Formula: see text] can be realized in the above manner.