Interval-valued Probability Research Articles

CT-Integral on Interval-Valued Sugeno Probability Measure and Its Application in Multi-Criteria Decision-Making Problems

It is well known that the complexity of the decision-making environment frequently coexists with the diversity of linguistic information in the decision-making process. In order to solve this kind of uncertain multi-criteria decision-making problem, reasonable measures and integrals should be established. In this paper, the discrete expression of the CT-integral on the interval-valued Sugeno probability measure is proposed. The CT-integral is the Choquet integral when the t-norm is T(x,y)=xy in the CT-integral and is a pre-aggregation function. Then, the CT-integral on interval-valued Sugeno probability measure is applied to solve end-of-life (EOL) strategy in order to determine multi-criteria decision-making problems. Compared with the general Choquet integral, the method proposed in this paper significantly improves the calculation process, that is, the calculation is simpler and the amount of calculation is smaller. A case study was performed in order to validate the effectiveness of this conclusion.

Assessment of Human Reliability Under the Conditions of Uncertainty: SPAR-H Methodology Interpreted in Terms of Interval-Valued Probabilities

Abstract The well-known standardized plant analysis risk-human reliability (SPAR-H) methodology is widely used for analysis of human reliability in complex technological systems. It allows assessing the human error probability taking into account eight important groups of performance shaping factors. Application of this methodology to practical problems traditionally involves assumptions which are difficult to verify under the conditions of uncertainty. In particular, it introduces only two possible values of the nominal human error probabilities (for diagnosis and for actions) which do not cover the whole spectrum of the tasks within operator's activity. In addition, although the traditional methodology considers the probabilities of human errors as the random variables, it operates only on a single predefined type of distribution for these variables and does not deal with the real situations in which the type of distribution remains uncertain. The paper proposes modification to the classical approach to enable more adequate modeling of real situations with the lack of available information. The authors suggest usage of the interval-valued probability technique and of the expert judgment on the maximum probability density for actual probabilities of human errors. Such methodology allows obtaining generic results that are valid for the entire set of possible distributions (not only for one of them). The modified methodology gives possibility to derive final assessments of human reliability in interval form indicating “the best case” and “the worst case.” A few numerical examples illustrate the main stages of the suggested procedure.

Teaching only about Keynes’s Contributions in Chapter 1 (Plus Possibly Chapters 2, 3, 6 and 26, if Time Permits) of the A Treatise on Probability Falls Far ,far Short of Providing Students with An Informed Foundation about Keynes’s Analysis of Decisionmaking,rationality and expectations

Teaching students about Keynes’s views on decision making ,expectations and rationality requires that the students have had a general introduction and overview of Parts II(interval valued probability), III(finite probabilities)and V(statistics, inexact measurement,and approximation, as discussed by Keynes in Chapter Four of the General Theory on pp.39-40 and 43-44) of Keynes’s A Treatise on Probability (1921) and a detailed coverage of chapters 15 and 17 of his Part II of the A Treatise on Probability on the importance of interval valued probability . Keynes argued in Chapter 15 that decision makers in the real world mainly rely on and use interval valued probability in a heuristic manner. They are not relying on the use of ordinal probability which is simply far, far too weak to provide a foundation for rationality. Meeks’ coverage of chapters 1-4 of Keynes’s A Treatise on Probability plus her possible excursion into chapters 6 and 26 would fail to provide the basic foundation needed by students to understand Keynes’s views in the Treatise on Probability and General Theory. Any primary source of discussion about the links between the A Treatise on Probability and General Theory MUST incorporate Keynes’s exchanges between himself and Hugh Townshend in 1937 and 1938,followed by the conflict between Tinbergen and Keynes in 1938-40 over Keynes’s imprecise probability(the intervals of Part II of the A Treatise on Probability ) using approximation and inexact measurement and Tinbergen’s approach based on precise probability and exact measurement .Meek’s coverage of these sources is simply nonexistent. Most importantly, Meeks needs to avoid the 1973 ,volume 8 version of the A Treatise on Probability in the CWJMK like the plague, as its editorial foreword by R. Braithwaite is completely flawed and erroneous.

Keynes Had No ‘Hidden Method’ in the A Treatise on Probability (1921): Keynes's Method Is an Explicit Inductive Logic Built on Inexact Measurement and Approximation, Which Was Openly Based on Boole’s Non Linear, Non Additive Approach Using Interval Values Probability

J M Keynes’s method was explicitly introduced and used in the A Treatise on Probability in Parts II, III and V. Keynes’s method is an inductive logic built on the mathematical logic and algebra of George Boole. Boole introduced non linearity and non additivity into his approach using interval valued probability that used lower and upper bounds. Boole’s approach, like Keynes’s, deals explicitly with problems like non comparability, non measurability and incommensurability that can’t be dealt with by additive and linear probability representations. Keynes initially introduced a brief discussion of these problems in chapter III of the A Treatise on Probability and Chapter 4 of the General Theory. Keynes called this method inexact measurement and approximation in chapter 15 of the A Treatise on Probability. It is impossible for Keynes to be anti-mathematical,anti-formalist, anti logicist, or a rationalist, given that, building on Boole, he created an inductive logic. Rationalists, by definition, do not accept the concept of induction, which is why they are called deductivists and not inductivists. Keynes, of course, took certain elements of the rationalist perspective and combined these elements (a priori thought, intuition, etc.) with certain elements of empiricism to create an early version of logical empiricism. Similarly, it is impossible for Keynes to have been anti logicist, anti formalist and anti mathematical because his work in Parts II and V of the A Treatise on Probability is logical,formal,and mathematical analysis built directly on the mathematical logic and algebra of George Boole. The belief that Keynes was a rationalist and/or anti logicist, anti formalist and anti mathematical is maintained by economists, such as A. Carabelli and R. O’Donnell, who never read or understood Parts II, III and V of the A Treatise on Probability or chapters 16-21 of G Boole’s 1854 The Laws of Thought. It is impossible for Keynes to be a rationalist and proclaim at the end of chapter 26 of the A Treatise on Probability that probability is the guide to life since all rationalists reject inductive logic.

An imprecise deep forest for classification

An imprecise deep forest classifier, which can be regarded as a modification of the deep forest proposed by Zhou and Feng, is presented in the paper. In the proposed classifier, the precise class probabilities at leaf nodes of decision trees in the deep forest are replaced with interval-valued probabilities produced by Walley’s imprecise Dirichlet model. The obtained sets of probabilities are averaged over all trees in a random forest as the imprecise probability masses. In order to use the stacking algorithm in the deep forest, a number of the class probability distributions are generated from sets of the random forest class probabilities such that decision trees of the next level of the deep forest cascade are trained on the basis of the extended training set which is added by new generated class distributions. Numerical examples illustrate how the incorporated imprecision may lead to outperforming results, especially, when the training set is rather small. The proposed classifier is the first modification of the deep forest, which efficiently deals with small datasets and takes into account a lack of sufficient training data. The ideas underlying the classifier may be a basis for development of a set on new machine learning models for small datasets.

A new approach to interval-valued probability measures, a formal method for consolidating the languages of information deficiency: Foundations

This article proposes a new formal definition of an interval valued probability measure (IVPM) based on a measure theoretic foundation, and shows that various uncertainties that occur in data associated in mathematical analyses, for example, in optimization under uncertainty models, can be unified and formulated within this one common IVPM framework facilitating the solution of many mathematical problems. This article develops a theory, we call generalized uncertainty theory, that will be characterized by the generation of upper and lower bounding functions enclosing all distributions that are possible from the given partial information.

Given Keynes's Definitions of Logical Probability and Evidential Weight, It Is Impossible for Keynes’s Approach to Measurement to Be an Ordinal Theory; His 'Non Numerical' Probabilities Must Be Based on Inexact and Imprecise Measurement Using Approximation Involving Boolean Upper and Lower Probability Bounds and Bounded Outcomes

Keynes spent all of his introductory Chapter One of the A Treatise on Probability emphasizing that his logical relation of probability (a relation of similarity – dissimilarity based on analogy) P(a/h)=α, where α came in degrees and 0≤α≤1. In chapter 6, Keynes introduced his logical relation of the evidential weight of the argument, V(a/h), but did not specify what it equaled, although he clearly stated that he would integrate both probability and weight together later. In chapter 26, Keynes specified that V(a/h)= w, where we came in degrees and 0≤w≤1. Keynes showed how to integrate both probability and weight together in his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. Probability and weight were also integrated together by Keynes using the theory of inexact measurement or approximation based on intervals with upper and lower bounds or limits. Ordinal probability can do none of these things. Advocates of the claim that Keynes’s theory was an ordinal theory of probability concentrate exclusively on a misinterpreted diagram that appears near the end of chapter 3 of the A Treatise on Probability. Thus, while Keynes’s theory can easily deal with ordinal probability with the aid of the Principle of Indifference, it is not part of Keynes’s main or major analysis in the A Treatise on Probability, which was interval probability. The same conclusion holds for numerical probability and rational expectations. Rational expectations are a very special case that occurs if w=1. The heterodox, institutionalist, or Post Keynesian approaches, involving irreducible, fundamental, or radical uncertainty, are a very, very special case that occurs only if w=0 in the very long run. Their argument, that measurement is not possible at all, be it exact or inexact, is rejected by Keynes in chapter 4 of the General Theory. Thus, while Keynes’s theory can easily deal with ordinal probability, it is impossible for the ordinal probability to deal with inexact and imprecise interval-valued probability measurement. Keynes’s theory of measurement involves interval-valued probability in the A Treatise on Probability with the numerical and ordinal probability being very special cases of his general theory of interval-valued probability. Keynes’s approach to measurement in the General Theory also involved inexact and imprecise measurement as deployed by Keynes in chapter 4 of the General Theory. No ordinal probability approach to measurement is used by Keynes in the General Theory. The great value of the 1936-1938 Keynes-Townshend exchanges is that they show that Keynes presented an approach to measurement that was imprecise and inexact in both the A Treatise on Probability and the General Theory.

Daniel Ellsberg on J.M. Keynes and F.H. Knight: risk ambiguity and uncertainty

This paper aims to critically evaluate the life and work of Daniel Ellsberg, with an intensive discussion on their relation to J.M. Keynes and F.H. Knight, the two great pioneers of the economics of uncertainty. Ellsberg seems to be a man in paradox. Although his remarkable paper on ambiguity and decision was published in a prestigious journal in 1962, it had not received due attention for a long time. Because he was so preoccupied in the 1960s and onward by letting the general public know the Pentagon papers, he could virtually have no time left to engage in purely academic activities. In the 21st century, however, the times have changed in favor of Ellsberg: we can see the dramatic return of interest in decision making under ambiguity. In this paper, we first deal with uncertainties that are not risks. A focal point of discussion will be the similarity and difference between Keynes and Knight. Kenneth Arrow’s skepticism about Knight on uncertainty will also be paid due attention. We will then turn to the main part of this paper, that is, the concept of ambiguity that was first introduced by Ellsberg. Both the two-color and the three-color problems will systematically be examined by help of numerical representations. While there are a variety of ways including the Nishimura–Ozaki approach for the resolution of the so-called Ellsberg paradox, the Keynesian approach by means of interval-valued probabilities, which was recently revived by Brady and applied by Sakai in many ways, is very simple and highly effective. In our opinion, the most amazing Ellsberg paradox lies in the fact that an accomplished economist specialized in the aversion of risk and uncertainty has dared to make a personal choice to risk everything such as degrading his social status and putting him in prison for a long period. Surely, the intellectual legacy of Ellsberg seems to be an intriguing research in paradox.

A novel approach to transformer fault diagnosis using IDM and naive credal classifier

Transformer fault diagnosis is important for improving the operational reliability of power systems. Despite great efforts to enhance the accuracy of fault diagnosis, the precise detection of transformer faults remains difficult. The major barriers are attributed to the lack of transformer fault records and related data, which may produce unconvincing diagnostic results, and to the fact that multiple latent faults in a transformer are difficult to detect distinctly. In this paper, an imprecise probability-based approach to transformer fault diagnosis by using the imprecise Dirichlet model and naive credal classifier is proposed. Instead of providing a single-valued probability, the approach calculates the interval probability for each possible type of transformer faults to explicitly indicate the uncertainty of the diagnosis caused by the insufficiency of historical fault records. Meanwhile, according to whether an overlap exists among the probability intervals of different types of faults, the proposed approach can output either one explicit fault or multiple probable latent faults that occurred in the transformer. In the proposed approach, the imprecise Dirichlet model (IDM) is used to evaluate the imprecise probabilistic relationships between each type of transformer fault and its corresponding symptom attributes. A naive credal classifier (NCC) is established to integrate the IDM estimation results and calculate the interval-valued probabilities of all types of possible faults. Then, the transformer fault diagnosis results can be obtained by the classification criteria of the NCC. The approach can compensate for the defects of conventional methods, including their inability to perform reliable diagnosis with insufficient fault samples or distinctly indicate the multiple latent transformer faults. The accuracy of diagnosis and the efficiency of transformer maintenance can be significantly improved by implementing the proposed approach. The effectiveness of the approach is verified through case studies.

Uncertainty and sensitivity analysis of energy assessment for office buildings based on Dempster-Shafer theory

Uncertainty and sensitivity analysis of building energy has become an active research area in order to consider variations of input variables and identify key variables influencing building energy. When there is only limited information available for uncertainty of building inputs, a specific probability for a given variable cannot be defined. Then, it is necessary to develop alternative approaches to probabilistic uncertainty and sensitivity analysis for building energy. Therefore, this paper explores the application of the Dempster-Shafer theory (DST) of evidence to conduct uncertainty and sensitivity analysis for buildings. The DST method is one of imprecise probability theories to allow combining uncertainty from different sources in terms of interval-valued probabilities in order to construct the belief and plausibility (two uncertainty measures) of system responses. The results indicate that the DST uncertainty analysis in combination with machine learning methods can provide fast and reliable information on uncertainty of building energy. It is recommended that at least two inherently different learning algorithms should be applied to provide robust simulation results of building energy. A spectrum of distributions should be implemented in global sensitivity analysis with the DST method because there are no specific distributions for intervals of input factors. Moreover, the stability of results from uncertainty and sensitivity analysis should be assessed when applying the DST method in building energy analysis.

Interval valued entropies for Dempster–Shafer structures

We first describe the basic ideas associated with the Dempster–Shafer belief structure and note its role in modeling uncertainty that manifests both imprecision and randomness. The inherent imprecision requires the use of intervals in expressing the associated probabilities. The ideas of plausibility and belief are introduced to provide upper and lower bounds on these interval-valued probabilities. We turn to the question of formalizing the entropy for these belief structures. As was the case with the associated probabilities the imprecision mandates the use of an interval-valued formulations for entropy. We suggest both Shannon type and Gini type formulations of interval-valued entropy for D–S belief structures. Here we make use of the classic scalar formulations of the D–S entropy introduced by Hoehle and Yager to provide the bounds for this interval-valued formulation. We discuss a more general type of belief structure in which our underlying uncertainty, while still manifesting imprecision, is based on a measure instead of a probability distribution. We show how to formulate entropy for these measure-based belief structures.

Keynes (Boole), Knight, (and Ellsberg) Against Ramsey, De Finetti, and Savage: It Had Nothing to Do with 'Radical Uncertainty', But Involved the Issue of Inexact, Indeterminate, Imprecise, Interval Valued (Weak Evidence) Probability Versus Exact, Determinate, Precise (Strong Evidence) Mathematical Probability

Recent articles in July, 2016 by P. Krugman and J. Fels reviewing Mervyn King's The End of Alchemy: Money, Banking, and the Future of the Global Economy, demonstrate a confusion by all three regarding what the issue was in the intellectual conflict between Keynes and Ramsey, between Ellsberg and de Finetti-Savage, and between Keynes (Boole), Knight, (and Ellsberg) and Ramsey, de Finetti, and Savage. The major issue had nothing to do with the alleged unmeasurability of so called quantitative probability, but with the issue of strong versus weak evidence. Keynes argued in both the A treatise on Probability and the General Theory that exact, precise probabilities could rarely be used in the real world of economic, business, and financial decision making because of the paucity and/or flimsiness of relevant available. Instead, non numerical probabilities had be used which incorporated the fact of what Keynes called major of the evidence deficiencies in the data, information, knowledge or evidence. This lead to Keynes's indeterminate, imprecise non numerical, interval valued probabilities or his conventional coefficient of risk and weight decision, c. Keynes's definition of uncertainty in the General Theory has absolutely nothing to do with radical uncertainty, but is defined on page 148 of the General Theory to be an inverse function of the weight of the evidence, w, where V(a/h)=w. V(a/h) is the evidential weight of the argument from h to conclusion a and 0≤w≤1. Radical uncertainty can occur only if w=0. That means that h=0 and there is no V relation that holds for conclusion a. It is Joan Robinson, GLS Shackle and Paul Davidson, not Keynes and Knight, who make this argument. King, Krugman and Fels are basing their analysis of uncertainty, not on what Knight and Keynes said, but on the false claims made by Robinson, Shackle and Davidson about Keynes and Knight. Knight's third category, estimates, is an inferior version of Keynes's c coefficient, which incorporated w, the weight of the evidence. Deficiencies in w lead to distorted or slanted additive, linear non numerical, decision weights or c coefficients. Knight was not able to integrate a concept of weight of the into his estimates category. However, he clearly demonstrated that a probability of ½, based on strong evidence, is to be preferred to an estimate of 1/2, which has the same form as a probability, but is based on weak evidence. This, of course, is clearly brought out in the first Ellsberg urn thought experiment and by Keynes in chapter 6 of the TP on pages 75-76. The Keynes-Townshend correspondence of 1937-38 proves beyond a shadow of a doubt that Keynes's General Theory is based on his probabilities and weights of the evidence. Ellsberg's conflict with de Finetti and Savage dealt with the same issues that separated Keynes from Ramsey. Ramsey never grasped the fact that interval valued probability CAN NEVER satisfy the additivity and linearity axioms of the mathematical calculus of probability.

Quantum interval-valued probability: Contextuality and the Born rule

We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not only can we quantify the idea of finite-precision measurement within our theory, but we can also suggest a possible resolution of the Meyer-Mermin debate on the impact of finite-precision measurement on the Kochen-Specker theorem.

Thomas Aquinas (A.K.A. the Medieval Church, Canonists, Scholastics, and Schoolmen) and J M Keynes on the Necessity of Excluding Smithhs Projectors, Imprudent Risk Takers, and Prodigals From Receiving Any Bank Loans

J. M. Keynes had already recognized in his 1930 A Treatise on Money that the major cause of deflation and inflation problems at the macro level was the ability of Smith’s Projectors, Imprudent risk takers, and Prodigals, whom Keynes categorized as Wall Street connected speculators and rentiers (modern day hedge funds, private equity firms, and investment banks) in his General Theory in 1936, were the fundamental problem in a capitalist society because they diverted bank loans from investment in durable capital formation into the financial manipulation of paper claims in various stock and money markets. The General Theory makes it even clearer because it reiterates what he had said in both the A Treatise on Money and in the 1932 Symposium on Usury in the Economic Journal in response to Keynes’s decision to publish a 1931 article by Henry Somerville. Keynes heavily supported Somerville’s major point about the dangers of using bank loans to finance speculative debt leverage practices versus loans to finance the purchase of durable capital equipment. This is the same point made by Smith with respect to loans made to his three categories of upper-income class citizens that Smith labeled as Projectors, Imprudent risk takers, and Prodigals. Both Smith and Keynes were influenced by Aristotle. Keynes admitted that he felt that the arguments of the “ancients”, “Medieval Church”, and “Schoolmen” were also influential in his general conclusions, although he differed from them in some specifics. Smith also used the term “Scholastics” and “Schoolmen” several times in his The Theory of Moral Sentiments. By far the most prominent member of the Medieval Church, Canonists, Scholastics, and Schoolmen was Thomas Aquinas. Somerville, in his June 1936 article in The Commonweal, titled, Usury as a New Issue, argued for a connection between the views of Keynes and Aquinas specifically. He had originally pointed out in the December 1931 issue of the Economic Journal Keynes’s connection with the “canonists” in his A Treatise on Money. In March 1932, the Savings and Usury Symposium was held in the Economic Journal. Keynes essentially states what he latter restated in the General Theory in chapter 23 in 1936. Keynes’s view on the dangers to the macroeconomy of making loans to speculators and rentiers did not change from 1930 until the end of his life. Aquinas’s work on decision making under conditions of ignorance, risk, uncertainty, imperfect information, asymmetric information and imprecise/indeterminate, interval-valued probabilities puts him heads and shoulders above any contemporary thinker until Adam Smith’s Wealth of Nations was published in 1776. Aquinas’s contribution has not been correctly appreciated because of his realization that precise, mathematical, point estimates of both prices and probabilities are generally impossible. This thus greatly constrains the use of the mathematical laws of the probability calculus upon which both neoclassical and modern economics is built on.

Interval Valued Generalized Stochastic Petri Nets for Imprecise Modeling in Unavailability Assessment

In this paper, we handle the issue of imprecision in evaluating the performance of the systems using Generalized Stochastic Petri Nets (GSPN). The basic probabilities generally considered in this type of nets are replaced by an interval-valued probability. This allows analysts to express their uncertainties related to the parameters of the different elements of the system and to assess the effect of this uncertainty on industrial system unavailability. Results obtained from varying the unavailability give some insights into the performance of the studied system. It is demonstrated how the imprecision led to important changes in the performance of the system. The used approach combines Generalized Stochastic Petri Nets (GSPN) and interval method to validate the obtained results obtained applied in safety systems.

Selection of risk reduction portfolios under interval-valued probabilities

A central problem in risk management is that of identifying the optimal combination (or portfolio) of improvements that enhance the reliability of the system most through reducing failure event probabilities, subject to the availability of resources. This optimal portfolio can be sensitive with regard to epistemic uncertainties about the failure events' probabilities. In this paper, we develop an optimization model to support the allocation of resources to improvements that mitigate risks in coherent systems in which interval-valued probabilities defined by lower and upper bounds are employed to capture epistemic uncertainties. Decision recommendations are based on portfolio dominance: a resource allocation portfolio is dominated if there exists another portfolio that improves system reliability (i) at least as much for all feasible failure probabilities and (ii) strictly more for some feasible probabilities. Based on non-dominated portfolios, recommendations about improvements to implement are derived by inspecting in how many non-dominated portfolios a given improvement is contained. We present an exact method for computing the non-dominated portfolios. We also present an approximate method that simplifies the reliability function using total order interactions so that larger problem instances can be solved with reasonable computational effort.

A Refutation of the Paul Davidson and Post Keynesian Claim that Paul Samuelson Was Never a Keynesian

A Refutation of the Paul Davidson and Post Keynesian Claim that Paul Samuelson Was Never a Keynesian

J M Keynes's Logical Theory of Probability Is Based on Interval Valued Probability, Not Ordinal Probability: How the Keynesian Fundamentalists Runde, Skidelsky, Carabelli, OODonnell, Meeks Got It All Wrong

The misbelief that Keynes’s logical theory of probability was based on a foundation of ordinal probability can be traced back to Frank Ramsey’s query about Keynes’s “mysterious non numerical probabilities.” Ramsey’ s flawed analysis was then passed on to a host of other philosophers and economists. Two of these were Robert Skidelsky and Gay Meeks. Their work, based on Ramsey, was then passed on to a host of economists. Neither Skidelsky not Meeks had any formal training in mathematics, logic, statistics, or probability. They simply read into certain parts of one chapter of Keynes’s A Treatise on Probability (1921), chapter 3, a gross misinterpretation based on the erroneous work of Ramsey. They confused unknown probabilities with indeterminate (interval valued) probabilities. A Huge amount of academic work, numbering in the thousands of articles and hundreds of books, has been published based on this confusion. The intellectual foundation for their flawed analysis rests on the uncritical acceptance of two extremely poor book reviews of the A Treatise on Probability made by F P Ramsey in 1922 and 1926 when he was 18 and 22 years old, respectively. The vastly superior reviews of Keynes’s book, made, for instance, by Francis Y Edgeworth, C. D Broad, and Bertrand Russell, were simply ignored. This paper shows how this error occurred and why.

Implementing Second-Order Decision Analysis: Concepts, Algorithms, and Tool

We present implemented concepts and algorithms for a simulation approach to decision evaluation with second-order belief distributions in a common framework for interval decision analysis. The rationale behind this work is that decision analysis with interval-valued probabilities and utilities may lead to overlapping expected utility intervals yielding difficulties in discriminating between alternatives. By allowing for second-order belief distributions over interval-valued utility and probability statements these difficulties may not only be remedied but will also allow for decision evaluation concepts and techniques providing additional insight into a decision problem. The approach is based upon sets of linear constraints together with generation of random probability distributions and utility values from implicitly stated uniform second-order belief distributions over the polytopes given from the constraints. The result is an interactive method for decision evaluation with second-order belief distributions, complementing earlier methods for decision evaluation with interval-valued probabilities and utilities. The method has been implemented for trial use in a user oriented decision analysis software.

Verified stochastic methods in geographic information system applications with uncertainty

Modern localization techniques are based on the Global Positioning System (GPS). In general, the accuracy of the measurement depends on various uncertain parameters. In addition, despite its relevance, a number of localization approaches fail to consider the modeling of uncertainty in geographic information system (GIS) applications. This paper describes a new verified method for uncertain (GPS) localization for use in GPS and GIS application scenarios based on Dempster-Shafer theory (DST), with two-dimensional and interval-valued basic probability assignments. The main benefit our approach offers for GIS applications is a workflow concept using DST-based models that are embedded into an ontology-based semantic querying mechanism accompanied by 3D visualization techniques. This workflow provides interactive means of querying uncertain GIS models semantically and provides visual feedback.