Axioms Of Probability Research Articles

On the cluster of the families of hybrid polynomial kernels in kernel density estimation

This study introduces a novel cluster of hybrid polynomial kernel families, designed to achieve significantly lower asymptotic mean integrated squared error compared to traditional kernels. These hybrid kernels are developed by heuristically combining classical polynomial kernels using probability axioms. An in-depth analysis of error propagation within these kernels is conducted, utilizing both simulation experiments and real-life datasets, including the Life Span of Batteries and COVID-19 datasets. The findings consistently demonstrate that the proposed hybrid kernels outperform their classical counterparts in various density estimation tasks across different distribution types and sample sizes. This research highlights the potential of hybrid polynomial kernels to enhance accuracy in density estimation, advocating for their adoption in statistical modelling and analysis.

A Probabilistic Evaluation of Surface Loading and Concentration as Metrics for Post Structural Fire Assessment Soot Sampling Data

Surface sampling and laboratory analysis for soot/combustion particulates was conducted following a fire at an education/research facility in the southwest United States. This provided a bank of data by which to probabilistically evaluate the behavior of soot loading (counts/mm2) and relative soot concentration (percent ratio; %R) as useful metrics for quantifying differences in soot impact across a building. Surface tape sampling and analysis via light microscopy were conducted via industry standard protocols, and resulting data from various building zones were selected to construct various comparisons. The performance of counts/mm2 and %R as metrics to identify differences in soot impact for each comparison was assessed by comparing inference generated by traditional Student’s t test, Mann Whitney U rank comparison (MW), and the directly calculated axiomatic probability associated with difference in detection (pΔfd). The fourteen (14) comparisons in which a significant difference was inferred via pΔfd was similarly indicated via Student’s t and/or MW in only four (4) instances. Further, approximately one half of the comparisons generated different inference via pΔfd for counts/mm2 and %R, with the former demonstrating better discriminatory ability. In broad view, the heuristic concept of comparing numerical “soot levels” (e.g., average) by either metric was not generally suitable for the distribution of the data. In contrast, pΔfd avoids the statistical bias imposed by traditional statistical inference, and ultimately the efficacy of post fire comparative surface sampling is as dependent upon the metric and inference model utilized as it is on the sampling and laboratory analytical protocols.

Primary-school children’s knowledge of transitivity of probability and of bounds of probability

ABSTRACT Primary-school children aged between 7 and 10 years were questioned about their conceptual knowledge of the first two axioms of qualitative probability. Children were questioned with prespecified examples and it was subsequently evaluated whether their answers were consistent with axioms. In addition, children’s knowledge of qualitative probability was investigated with universal and existential statements about axioms. Children showed good knowledge of probability with prespecified examples. In contrast, majority showed poor knowledge when asked about some of the universal and existential statements. An age-related improvement was not detected. In particular, only 25% said that the transitivity was necessary, i.e. that it applied to all events. 60% said that an event may be both possible and impossible. Only 42% said that all impossible events were equally likely and 54% said that all certain events were equally likely. Only 56% said that probability has both upper and lower bounds.

Validation of local approach‐based statistical models for cleavage fracture of ferritic steels in uniaxial tension

AbstractFive major local approach (LA)‐based statistical cleavage fracture models are evaluated in uniaxial tension. The focus is to inspect their mathematical conformity to the axiom of probability on normality and their physical adherence to the consensus that cleavage fracture occurs after plastic deformation. Only the LA model that incorporates a stress‐state and temperature dependent threshold satisfies both requirements. Two sets of experimental data in uniaxial tension are analyzed to support the arguments.

The Philosophy of Probability Values Behaviour, through Fractions and Composite Probability Function in the Continuous Case

This paper deals with probability theory, and it is an extension to a published paper that has the same title, but for the discrete case. This present paper is aiming to study probability values behavior, in the case of continuous sample space, through fractions intervals and composite function. This aim tends to study the value behavior rather than finding the value itself. Also, this aim requires usage of some concepts of continuity, geometric probability, and measure theory, which also need a brief treatment. This paper is mainly using an experiment with a design that helps to study the probability fractions values in the form of intervals in the case of one direction movement and in the case of different directions. As a result, every case reflects some aspects of probability values behavior and can clarify many important characteristics of the probability theory. In addition to applying the composite function by some important theorems of conditional probability. These are besides a proven proposition that helps to design experiment, upon the understanding of the case nature. In addition to a corollary that allows to visualize negative probability values, as a particular case (trial), that upon the validity of the explanation of negativity which should be consistent with the probability axioms.

Explanatory virtues and reasons for belief

Abstract In this essay, I address an objection to inference to the best explanation due to Bas C. van Fraassen, according to which explanatory virtues cannot confirm a theory, since they make the theory more informative and thus less likely to be true given the probability axioms. I try to show that van Fraassen’s argument, once made precise, is deductively invalid, and that even an ampliative version of the argument (i) implies, absurdly, that no theory is confirmed by its fit with empirical data; (ii) fails to account for confirmatory closure under deduction; and (iii) falsely presupposes that a theory and its sub-theories can be competing explanations.

The Philosophy of Probability Values Behaviour through Fractions and Composite Probability Function for Independent Events in the Discrete Case

This paper focuses on dealing with probability theory, and it adopted an objectivism philosophic approach, as well as a mathematical approach. These approaches aim to study the probability values behaviour as discrete quantities in the case of discrete sample space for independent events through fractions and composite functions. This requires a discussion of the usage of probability statements, the causes behind the existence of probabilistic phenomenon, and an explanation that admits to measuring causality in probability theory through the concept of complement and fractions. The paper uses an experiment with a design that addresses the shortcoming of traditional experiments through the concept of fractions. This, in turn, reflects some aspects of the probabilistic behaviour, including some important consequences that follow through. This paper also uses the relative frequency of events and the probability axioms, which provides the sample space to include all possible events in the form of sequences and sub sequences. The paper further provides some definitions that define some elements of the fractions probabilities and the sample space, alongside some proven propositions, lemma, and corollaries that admit to calculating composite probability functions. This is in addition to a brief discussion of the continuous cases.

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

NECESSARY AND SUFFICIENT CONDITIONS FOR DOMINATION RESULTS FOR PROPER SCORING RULES

AbstractScoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms of the probability calculus, but Predd et al. [9] have shown that given a finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal.

Extending Kolmogorov's Axioms for a Generalized Probability Theory on Collections of Contexts.

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem.

Are There Statistically Impenetrable Contexts?

Bayesians claim that reason is conducted according to the axioms of probability. In addition, to this, they claim that Bayesian methodology gives us an account of the scientific method―an account that describes the actual practice of science. These are two claims. Bayesian decision theory falls short of its descriptivist claims due to psychological, contextual, and qualitative shortcomings. In some contexts, there may be problems with prediction and prior probability. Even though there might be an acceptable conclusion to a problem, Bayes’ cannot even describe it. I use the example of a disjunction, wherein we cannot assign prior probabilities to either disjunct due to qualitative differences. Bayes only deals with degrees of personal belief. Here we run into not only the problem of assigning prior probability but also problems of uncertain evidence, and principles of relational acceptance. Not the descriptive aspect in jeopardy, but also the Bayesian conceptualization of the scientific method.

Accuracy, probabilism, and the insufficiency of the alethic

The best and most popular argument for probabilism is the accuracy-dominance argument, which purports to show that alethic considerations alone support the view that an agent’s degrees of belief should always obey the axioms of probability. I argue that extant versions of the accuracy-dominance argument face a problem. In order for the mathematics of the argument to function as advertised, we must assume that every omniscient credence function is classically consistent; there can be no worlds in the set of dominance-relevant worlds that obey some other logic. This restriction cannot be motivated on alethic grounds unless we’re also willing to accept that rationality requires belief in every metaphysical necessity, as the distinction between a priori logical necessities and a posteriori metaphysical ones is not an alethic distinction. To justify the restriction to classically consistent worlds, non-alethic motivation is required. And thus, if there is a version of the accuracy-dominance argument in support of probabilism, it isn’t one that is grounded in alethic considerations alone.

The Maxim of Probabilism, with special regard to Reichenbach

It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations.

Deciding Koopman's qualitative probability

In their recent paper in this journal, Delgrande, Renne and Sack study qualitative probability and discuss its role in Artificial Intelligence. Building on related work by de Finetti, Scott, Segerberg and others, the authors provide general axioms for qualitative probability. In this paper we investigate Koopman's conditional qualitative probability from the computational viewpoint. An important part of his work, published in the Annals of Mathematics in 1940-1941, deals with finite conjunctions K of statements of the form “the probability of a given h does not exceed the probability of b given k”, with a,b,h,k elements of a boolean algebra. Upon coding these elements by boolean formulas, we provide a decision procedure to check the consistency of any such K. As an immediate consequence, also inferences in Koopman's probability theory are shown to be computable. These problems of qualitative probability theory significantly generalize Boole's (typically quantitative) problem of estimating the possible probabilities of a new event given the probabilities of other events. Boole's classical problem today is known as the optimization version of the probabilistic satisfiability problem PSAT. In 1986 Nilsson published an influential paper on this subject in this journal. The scope of our results is much larger than that of PSAT, because Koopman's conjunctions K also formalize the key notion of independence. Some familiarity with boolean logic and finite boolean algebras is the only prerequisite for this paper.

Deep memory distortions

Deep distortions are a new family of memory biases that comprise one of the two basic varieties of false memory. The first and older variety, surface distortions, are specific item or source memories that are erroneous because the events did not happen. The new variety, deep distortions, are emergent properties of multiple specific memories. They are relations among such memories that are false because they violate objective logical rules that real-world events must obey. I discuss four deep distortions for which substantial data have accumulated: overdistribution, super-overdistribution, non-additivity, and impossible conjunctions. These phenomena violate four axioms of classical probability (numerical bound, universal event, additivity, and countable additivity) and two rules that follow from them (empty set and monotonicity). Their psychological significance lies in four facts about them: (a) They demonstrate that although events in the real world are compensatory, our memories of them are not; (b) they establish that we persistently over remember experience; (c) they reveal that surface distortions are by-products of deep distortions; and (d) they pose the theoretical conundrum of how the structure of memory could so thoroughly misrepresent the objective structure of the events we are remembering.

Probability Axioms and Set Theory Paradoxes

In this paper, we show that Zermelo–Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo–Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov’s probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability—both based on simple notions of symmetry and independence.

Deep distortion

ABSTRACT Conventional false memories recount events that either did not happen (item errors) or that happened in a different context (source errors). Fuzzy-trace theory predicts deeper anomalies that lie behind conventional false memories. These deep distortions are structural irregularities in the ways that specific recountings are related to each other or to some objective standard (e.g., the 0 and 1 limits of probability). I discuss five deep distortions for which substantial data have accumulated: overdistribution, non-additivity, conjunction illusions, non-compensation, and super-overdistribution. Together, these phenomena violate the disjunction and additivity axioms of probability, as well as the law of the excluded middle. The theoretical problem they pose is to explain how valid representations of our experience produce memory regularities that violate our experience in the most fundamental ways.

An epidemiological index for drought vulnerability in the Rio Grande do Norte State, Brazil.

In the Northeast Brazil (NEB), the impacts of climate extreme events such as severe droughts are aggravated by poverty and poor socioeconomic conditions. In this region, such events usually result in the spread of endemic diseases, problems in water distribution, and agricultural losses, often leading to an increase in the population's vulnerability. Thus, this study aims to evaluate the microregions of the Rio Grande do Norte (RN) state, in the NEB, according to the Epidemiological Index for Drought Vulnerability (EIDV). We mapped and classified the microregions according to three dimensions of vulnerability: risk, susceptibility, and adaptive capacity. We also verified potential associations between drought risk and epidemiological vulnerability. The EIDV was calculated by considering the three dimensions of vulnerability as mutually exclusive events and applying the third axiom of probability. Then we carried out a cluster analysis in order to classify the microregions according to similarities in the EIDV. Odds ratio were also calculated in order to evaluate the odds of microregions having a high susceptibility to diseases and high vulnerability given the drought risk. Results showed that the Pau dos Ferros, Seridó Ocidental, Seridó Oriental, and Umarizal microregions were the most vulnerable, while Natal and Litoral Sul were the least vulnerable. Regarding the dimensions of vulnerability, we observed that almost the entire RN state exhibited high drought risk. Pau dos Ferros and Umarizal had the highest susceptibility and Litoral Nordeste presented the worst adaptive capacity to the effects of drought on health. The EIDV revealed that the population of the RN state needs improvements in living conditions and health, since socioeconomic status is one of the factors that most influence the vulnerability of microregions, which in turn is aggravated by drought risk.

The P–T Probability Framework for Semantic Communication, Falsification, Confirmation, and Bayesian Reasoning

Many researchers want to unify probability and logic by defining logical probability or probabilistic logic reasonably. This paper tries to unify statistics and logic so that we can use both statistical probability and logical probability at the same time. For this purpose, this paper proposes the P–T probability framework, which is assembled with Shannon’s statistical probability framework for communication, Kolmogorov’s probability axioms for logical probability, and Zadeh’s membership functions used as truth functions. Two kinds of probabilities are connected by an extended Bayes’ theorem, with which we can convert a likelihood function and a truth function from one to another. Hence, we can train truth functions (in logic) by sampling distributions (in statistics). This probability framework was developed in the author’s long-term studies on semantic information, statistical learning, and color vision. This paper first proposes the P–T probability framework and explains different probabilities in it by its applications to semantic information theory. Then, this framework and the semantic information methods are applied to statistical learning, statistical mechanics, hypothesis evaluation (including falsification), confirmation, and Bayesian reasoning. Theoretical applications illustrate the reasonability and practicability of this framework. This framework is helpful for interpretable AI. To interpret neural networks, we need further study.

Rebooting the new evidence scholarship

The new evidence scholarship addresses three distinct approaches: legal probabilism, Bayesian decision theory and relative plausibility theory. Each has major insights to offer, but none seems satisfactory as it stands. This paper proposes that relative plausibility theory be modified in two substantial ways. The first is by defining its key concept of plausibility, hitherto treated as primitive, by generalising the standard axioms of probability. The second is by complementing the descriptive component of the theory with a normative decision theory adapted to legal process. Because this version of decision theory is based on plausibilities rather than probabilities, it generates plausibilistic expectations as outputs. Because these outputs are comparable, they function as relative plausibilities. Hence the resulting framework is an extension of relative plausibility theory, but it retains deep ties to legal probabilism, through the proposed definition of plausibility, and to Bayesian decision theory, through the normative use of decision theory.