Traditional Probability Theory Research Articles

Research on dependent evidence combination based on principal component analysis.

Dempster-Shafer evidence theory, as a generalization of probability theory, is a powerful tool for dealing with a variety of uncertainties, such as incompleteness, ambiguity, and conflict. Because of its advantages in information fusion compared with traditional probability theory, it is widely used in various fields. However, the classic Dempster's combination rule assumes that evidences are independent of each other, which is difficult to satisfy in real life. Ignoring the dependence among the evidences will lead to unreasonable fusion results, and even wrong conclusions. Considering the limitations of D-S evidence theory, this paper proposed a new evidence fusion model based on principal component analysis (PCA) to deal with the dependence among evidences. First, the approximate independent principal components of each information source were obtained based on principal component analysis. Second, the principal component data set was used as a new information source for evidence theory. Third, the basic belief assignments (BBAs) were constructed. As the fundamental construct of evidence theory, a BBA is a probabilistic function corresponding to each hypothesis, quantifying the belief assigned based on the evidence at hand. This function facilitates the synthesis of disparate evidence sources into a mathematically coherent and unified belief structure. After constructing the BBAs, the BBAs were fused and a conclusion was drawn. The case study verified that the proposed method is more robust than several traditional methods and can deal with redundant information effectively to obtain more stable results.

A model of stochastic memoization and name generation in probabilistic programming: categorical semantics via monads on presheaf categories

Stochastic memoization is a higher-order construct of probabilistic programming languages that is key in Bayesian nonparametrics, a modular approach that allows us to extend models beyond their parametric limitations and compose them in an elegant and principled manner. Stochastic memoization is simple and useful in practice, but semantically elusive, particularly regarding dataflow transformations. As the naive implementation resorts to the state monad, which is not commutative, it is not clear if stochastic memoization preserves the dataflow property -- i.e., whether we can reorder the lines of a program without changing its semantics, provided the dataflow graph is preserved. In this paper, we give an operational and categorical semantics to stochastic memoization and name generation in the context of a minimal probabilistic programming language, for a restricted class of functions. Our contribution is a first model of stochastic memoization of constant Bernoulli functions with a non-enumerable type, which validates data flow transformations, bridging the gap between traditional probability theory and higher-order probability models. Our model uses a presheaf category and a novel probability monad on it.

The logic of uncertainty in law and life

Abstract My central interest is decision making in the presence of epistemic uncertainty. A method appropriate for both specialized inquiries and everyday reasoning is based on credal logic, which employs multivalent degrees of belief rather than traditional probability theory. It accounts for epistemic uncertainty as unallocated belief. It holds that, when facing real uncertainty, if a person believes a and believes b, then the person believes a and b together. This brand of multivalent logic underlies and justifies how legal decision makers and the rest of us find facts in a world infused with epistemic uncertainty. Indeed, this article closes by showing the equivalence of multivalent logic and inference to the best explanation. By demonstrating this similarity in reasoning, I aim to shore up our faith in the logic of traditional legal reasoning.

Logic in Law and Science

My central interest is decisionmaking in the presence of epistemic uncertainty. A method appropriate for both specialized inquiries and everyday reasoning is based on credal logic, which employs multivalent degrees of belief rather than traditional probability theory. It accounts for epistemic uncertainty as unallocated belief. It holds that, when facing real uncertainty, if a person believes a and believes b, then the person believes a and b together. This brand of multivalent logic underlies and justifies how legal decisionmakers and the rest of us find facts in a world infused with epistemic uncertainty. Indeed, this Article closes by showing the equivalence of multivalent logic and inference to the best explanation. By demonstrating this similarity in reasoning, I aim to shore up our faith in the logic of traditional legal reasoning.

A survey of decision making and optimization under uncertainty

Recent advances in decision making have incorporated both risk and ambiguity in decision theory and optimization methods. These methods implement a variety of uncertainty representations from probabilistic and non-probabilistic foundations, including traditional probability theory, sets of probability measures, uncertainty sets, ambiguity sets, possibility theory, evidence theory, fuzzy measures, and imprecise probability. The choice of uncertainty representation impacts the expressiveness and tractability of the decision models. We survey recent approaches for representing uncertainty in both decision making and optimization to clarify the trade-offs among the alternative representations. Robust and distributionally robust optimization are surveyed, with particular attention to standard form ambiguity sets. Applications of uncertainty and decision models are also reviewed, with a focus on recent optimization applications. These applications highlight common practices and potential research gaps. The intersection of behavioral decision making and robust optimization is a promising area for future research and there is also opportunity for further advances in distributionally robust optimization in sequential and multi-agent settings.

Using deep learning to preserve data confidentiality

Preserving data confidentiality is crucial when releasing microdata for public-use. There are a variety of proposed approaches; many of them are based on traditional probability theory and statistics. These approaches mainly focus on masking the original data. In practice, these masking techniques, despite covering part of the data, risk leaving sensitive data open to release. In this paper, we approach this problem using a deep learning-based generative model which generates simulation data to mask the original data. Generating simulation data that holds the same statistical characteristics as the raw data becomes the key idea and also the main challenge in this study. In particular, we explore the statistical similarities between the raw data and the generated data, given that the generated data and raw data are not obviously distinguishable. Two statistical evaluation metrics, Absolute Relative Residual Values and Hellinger Distance, are the evaluation methods we have decided upon to evaluate our results. We also conduct extensive experiments to validate our idea with two real-world datasets: the Census Dataset and the Environmental Dataset.

The silliness of magical realism

Relative plausibility, even after countless explanatory articles, remains an underdeveloped model bereft of underlying theory. Multivalent logic, a fully developed and accepted system of logic, comes to the same endpoint as relative plausibility. Multivalent logic would thus provide the missing theory, while it would resolve all the old problems of using traditional probability theory to explain the standards of proof as well as the new problems raised by the relative plausibility model. For example, multivalent logic resolves the infamous ‘conjunction paradox’ that traditional probability creates for itself, and which relative plausibility tries to sweep under the rug.Yet Professors Allen and Pardo dismiss multivalent logic as magical realism when applied to legal factfinding. They reject this ring buoy because they misunderstand nonclassical logic, as this response explains.

Common Sense on Standards of Proof

The law speaks clearly on the standards of proof, but listeners often misunderstand its words. This article tries, with some common sense and a modicum of multivalent logic, to explain how the law expects its standards to be applied, and then to show how the law thereby avoids such complications as the conjunction paradox. First, in accordance with belief function theory, the factfinder should start at zero belief. Given imperfect evidence, the factfinder will end up retaining a fair amount of uncommitted belief. As evidence comes in, though, the factfinder will form a belief in the truth of the disputed fact but also form a disbelief, or a belief in the fact’s falsity. At the close of evidence, the standard of proof requires only comparing belief and disbelief. For example, the civil standard, rather than asking whether a fact more likely than not happened according to traditional probability theory, asks whether the factfinder believes in the fact’s happening more than the factfinder believes that the fact did not happen. The burdened party need not push proof above 50% by dispelling the phantoms of every possibility, and the opponent need not generate a competing version of truth but can instead rely on denial to demand that the burdened party generate a belief. Second, belief and disbelief being nonadditive partial truths, the mathematical result is that one cannot combine beliefs by traditional probability theory, as by using the product rule designed for conjunction of betting odds. Instead, one should use fuzzy logic, including its rule that conjoined likelihood equals the likelihood of the least likely element. Linking the elements in a chain tells a story that is as likely as its weakest link. Consequently, if each element of a claim or defense passes the standard of proof, the conjunction of elements will pass the standard of proof. The conjunction paradox thus vaporizes for factfinding, just as the law has always maintained. The law found the way to decide in accord with our best knowledge of the facts.

Extended Importance Sampling for Reliability Analysis under Evidence Theory

In early engineering practice, the lack of data and information makes uncertainty difficult to deal with. However, evidence theory has been proposed to handle uncertainty with limited information as an alternative way to traditional probability theory. In this contribution, a simulation-based approach, called ‘Extended importance sampling’, is proposed based on evidence theory to handle problems with epistemic uncertainty. The proposed approach stems from the traditional importance sampling for reliability analysis under probability theory, and is developed to handle the problem with epistemic uncertainty. It first introduces a nominal instrumental probability density function (PDF) for every epistemic uncertainty variable, and thus an ‘equivalent’ reliability problem under probability theory is obtained. Then the samples of these variables are generated in a way of importance sampling. Based on these samples, the plausibility and belief (upper and lower bounds of probability) can be estimated. It is more efficient than direct Monte Carlo simulation. Numerical and engineering examples are given to illustrate the efficiency and feasible of the proposed approach.

Uncertainty quantification under hybrid structure of probability-fuzzy parameters in Gaussian plume model

In almost all the realistic circumstances, such as atmospheric dispersion and health risk assessment analysis, it is very essential to include all the information into modelling. The parameters associated with a particular model may include different kinds of variability, imprecision and uncertainty. More often, it is seen that available information is interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all available information, data or model parameters affected by variability, imprecision and uncertainty can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement error or data obtained from expert judgment or subjective interpretation of available data or information. Thus, model parameters’ data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent subjective uncertainty. Possibility theory is another branch of mathematics which is used as a tool to describe the parameters with insufficient knowledge. It may also happen that the probabilistic information about the parameters may be imprecise; that is, it might be associated with some probability bounds. Such types of uncertainty are incorporated into imprecise probability, a theory that extends traditional probability theory by allowing for intervals or sets of probabilities. In this paper, an attempt has been made to combine imprecise probability knowledge and possibility knowledge and draw the uncertainty. The article describes a combination of imprecise probability and fuzzy knowledge and applied to atmospheric dispersion with a case study. Further, fuzziness measure has been utilized in quantifying the uncertainty.

Assessing transition probability of bridge deterioration using Dempster–Shafer theory of evidence

Reliable deterioration models for forecasting the condition of bridge elements, systems, and networks are essential components of any bridge management system. Estimating deterioration transition probabilities is the basis of the popular Markov chain deterioration model. Transition probabilities ha ve been produced in the literature by statistical and probabilistic analysis performed on available bridge inspection and condition data. The available data may be limited or inconsistent with the Markov chain transition period. In addition, the conventional probability theory has limitations when applied to problems with a stochastic nature such as bridge deterioration. This paper introduces a novel method based on the theory of evidence for bridge deterioration modeling through expert judgment elicitation. The advantages of the theory of evidence over the traditional probability theory are discussed and the process for the theory implementation is demonstrated with a case study to validate the application of Dempster–Shafer theory of evidence to estimate the transition probabilities. Based on the results, the theory of evidence is proposed as a scientific expert judgement elicitation technique in the area of bridge condition rating and deterioration modeling. Expert judgment elicitation and theory of evidence application hold potential in the field of bridge management and require further investigation and research.

Empirical Data Analytics

In this paper, we propose an approach to data analysis, which is based entirely on the empirical observations of discrete data samples and the relative proximity of these points in the data space. At the core of the proposed new approach is the typicality—an empirically derived quantity that resembles probability. This nonparametric measure is a normalized form of the square centrality (centrality is a measure of closeness used in graph theory). It is also closely linked to the cumulative proximity and eccentricity (a measure of the tail of the distributions that is very useful for anomaly detection and analysis of extreme values). In this paper, we introduce and study two types of typicality, namely its local and global versions. The local typicality resembles the well-known probability density function (pdf), probability mass function, and fuzzy set membership but differs from all of them. The global typicality, on the other hand, resembles well-known histograms but also differs from them. A distinctive feature of the proposed new approach, empirical data analysis (EDA), is that it is not limited by restrictive impractical prior assumptions about the data generation model as the traditional probability theory and statistical learning approaches are. Moreover, it does not require an explicit and binary assumption of either randomness or determinism of the empirically observed data, their independence, or even their number (it can be as low as a couple of data samples). The typicality is considered as a fundamental quantity in the pattern analysis, which is derived directly from data and is stated in a discrete form in contrast to the traditional approach where a continuous pdf is assumed a priori and estimated from data afterward. The typicality introduced in this paper is free from the paradoxes of the pdf. Typicality is objectivist while the fuzzy sets and the belief-based branch of the probability theory are subjectivist. The local typicality is expressed in a closed analytical form and can be calculated recursively, thus, computationally very efficiently. The other nonparametric ensemble properties of the data introduced and studied in this paper, namely, the square centrality, cumulative proximity, and eccentricity, can also be updated recursively for various types of distance metrics. Finally, a new type of classifier called naive typicality-based EDA class is introduced, which is based on the newly introduced global typicality. This is only one of the wide range of possible applications of EDA including but not limited for anomaly detection, clustering, classification, control, prediction, control, rare events analysis, etc., which will be the subject of further research.

Cooperative grey games and an application on economic order quantity model

Purpose – The purpose of this paper is to extend the results of Meca et al. (2004) depending on the grey information revealed by the individual firms. Design/methodology/approach – The authors introduce cooperative grey games and focus on sharing ordering cost rule (SOC-rule) to distribute the joint cost. Findings – In this study, the authors introduce a model, where inventory costs are assumed as grey numbers instead of crisp or stochastic ones studied in literature. At first, grey numbers and classical cooperative inventory games are recalled. Then, cooperative grey games are introduced and related results are given. Finally, an application is performed for three shotgun companies in Turkey. Originality/value – It is an effective approach for theoretical analysis of systems with imprecise information and incomplete samples. Therefore, grey system theory, rather than the traditional probability theory and fuzzy set theory, is better suited to model the inventory problems by using cooperative game theory. To the best of the knowledge no study exists modeling inventory situations by using cooperative grey games. From this point of view this study is a pioneering work on a promising topic.

Outside the box: an alternative data analytics framework

In this paper, an alternative framework for data analytics is proposed which is based on the spatially-aware concepts of eccentricity and typicality which represent the density and proximity in the data space. This approach is statistical, but differs from the traditional probability theory which is frequentist in nature. It also differs from the belief and possibility-based approaches as well as from the deterministic first principles approaches, although it can be seen as deterministic in the sense that it provides exactly the same result for the same data. It also differs from the subjective expert-based approaches such as fuzzy sets. It can be used to detect anomalies, faults, form clusters, classes, predictive models, controllers. The main motivation for introducing the new typicality- and eccentricity-based data analytics (TEDA) is the fact that real processes which are of interest for data analytics, such as climate, economic and financial, electro-mechanical, biological, social and psychological etc., are often complex, uncertain and poorly known, but not purely random. Unlike, purely random processes, such as throwing dices, tossing coins, choosing coloured balls from bowls and other games, real life processes of interest do violate the main assumptions which the traditional probability theory requires. At the same time they are seldom deterministic (more precisely, have always uncertainty/noise component which is nondeterministic), creating expert and belief-based possibilistic models is cumbersome and subjective. Despite this, different groups of researchers and practitioners favour and do use one of the above approaches with probability theory being (perhaps) the most widely used one. The proposed new framework TEDA is a systematic methodology which does not require prior assumptions and can be used for development of a range of methods for anomalies and fault detection, image processing, clustering, classification, prediction, control, filtering, regression, etc. In this paper due to the space limitations, only few illustrative examples are provided aiming proof of concept.

Uncertainty modelling of atmospheric dispersion by stochastic response surface method under aleatory and epistemic uncertainties

The parameters associated to a environmental dispersion model may include different kinds of variability, imprecision and uncertainty. More often, it is seen that available information is interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all available information, data or model parameters affected by variability, imprecision and uncertainty, can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement error or data obtained from expert judgement or subjective interpretation of available data or information. Thus for model parameters, data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent subjective uncertainty. Possibility theory is used as a tool to describe parameters with insufficient knowledge. Based on the polynomial chaos expansion, stochastic response surface method has been utilized in this article for the uncertainty propagation of atmospheric dispersion model under consideration of both probabilistic and possibility information. The proposed method has been demonstrated through a hypothetical case study of atmospheric dispersion.

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellisʼs largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellisʼs reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.

Environmental Risk Modelling under Probability-normal Interval-valued Fuzzy Number

In almost all the realistic circumstances, such as health risk assessment and uncertainty analysis of atmospheric dispersion, it is very essential to include all the information into modelling. The parameters associated to a particular model may include different kind of variability, imprecision and uncertainty. More often, it is seen that available informations are interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all of available information, data or model parameters affected by variability, imprecision and uncertainty can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement errors or data obtained from expert judgement or subjective interpretation of available data or information. Thus, model parameters, data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent them. Possibility theory and fuzzy set theory is another branch of mathematics which is used as a tool to describe the parameters with insufficient or vague knowledge. In this paper, an attempt has been made to combine probability knowledge and possibility knowledge and draw the uncertainty. The paper describes an algorithm for combining probability distribution and interval-valued fuzzy number and applied to environmental risk modelling with a case study. The primary aim of this paper is to propagate the proposed method. Computer codes are prepared for the proposed method using MATLAB.

A fuzzy random model for rail freight car fleet sizing problem

In the area of freight transport, the railroads of almost all countries face with strong competition and a prominent trend of market reduction. It has become imperative for rail systems to develop better planned instruments for more rational and efficient utilization of freight cars that represent a great amount of total investments. In this paper a new formulation and a solution procedure is proposed for optimizing the fleet size and freight car allocation in the presence of uncertainty. The uncertainty of the rail freight car demand is often tackled from the traditional probability theory. However, various types of uncertainties and fuzziness are inherent in real rail freight transport operating environment. In this paper, the issue of rail freight car fleet sizing and allocation problem will be addressed under such circumstances. Specifically, an approach based on optimal control theory by considering the fuzziness and randomness for rail freight car demand is developed here. The problem is formulated as the problem of finding an optimal fuzzy regulator for a fuzzy linear system with fuzzy quadratic performance index and fuzzy random initial conditions. Numerical example is given to illustrate the model and solution methodology.

Epistemic Uncertainty Analysis: An Approach Using Expert Judgment and Evidential Credibility

When dealing with complex systems, all decision making occurs under some level of uncertainty. This is due to the physical attributes of the system being analyzed, the environment in which the system operates, and the individuals which operate the system. Techniques for decision making that rely on traditional probability theory have been extensively pursued to incorporate these inherent aleatory uncertainties. However, complex problems also typically include epistemic uncertainties that result from lack of knowledge. These problems are fundamentally different and cannot be addressed in the same fashion. In these instances, decision makers typically use subject matter expert judgment to assist in the analysis of uncertainty. The difficulty with expert analysis, however, is in assessing the accuracy of the expert's input. The credibility of different information can vary widely depending on the expert’s familiarity with the subject matter and their intentional (i.e., a preference for one alternative over another) and unintentional biases (heuristics, anchoring, etc.). This paper proposes the metric of evidential credibility to deal with this issue. The proposed approach is ultimately demonstrated on an example problem concerned with the estimation of aircraft maintenance times for the Turkish Air Force.

Impugning Randomness, Convincingly

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn’t particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenarios like the lottery one. We attempt to rectify that.