Bruno de Finetti and Fuzzy Probability Distributions

Journal of Mathematical Analysis and Applications, 1982

A general framework for a theory is presented that encompasses both statistical uncertainty. which falls within the province of probability theory, and nonstatistical uncertamty. which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. FUZZJ Sers I (1978). 3-281. The concept of a fuzzy integral ts used to define the expected value of a random vartable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence IS formulated, various convergence concepts are defined and their relationshtps are studied by using a Chebyshevlike inequality for the fuzzy Integral. The possibility of using this theory m Bayestan estimation with fuzzy prior mformation IS explored.

Advanced Fuzzy Logic Approaches in Engineering Science, 2019

Human health risk assessment is an important and a popular aid in the decision-making process. The basic objective of risk assessment is to assess the severity and likelihood of impairment to human health from exposure to a substance or activity that under plausible circumstances can cause harm to human health. One of the most important aspects of risk assessment is to accumulate knowledge on the features of each and every available data, information and model parameters involved in risk assessment. It is observed that most frequently model parameters, data, and information are tainted with aleatory and epistemic uncertainty. In such situations, fuzzy set theory or probability theory or Dempster-Shafer theory (DSS) can be explored to represent uncertainty. If all the three types of uncertainty coexist how far computation of the risk is concern, two ways to deal with the situation either transform all the uncertainties to one type of format or need for joint propagation of uncertaint...

2011

On two important counts, the Zadehian theory of fuzzy sets urgently needs to be restructured. First, it can be established that for a normal fuzzy number N = [α, β, γ] with membership function Ψ 1 (x), if α ≤ x ≤ β, Ψ 2 (x), if β ≤ x ≤ γ, and 0, otherwise, Ψ 1 (x) is in fact the distribution function of a random variable defined in the interval [α, β], while Ψ 2 (x) is the complementary distribution function of another random variable defined in the interval [β, γ]. In other words, every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. This is how a normal fuzzy number should be constructed, and this is how partial presence of an element in a fuzzy set has to be defined. Hence the measure theoretic matters with reference to fuzziness have to be studied accordingly. Secondly, the field theoretic matters related to fuzzy sets are required to be revised all over again because in the current definition of the complement of a fuzzy set, fuzzy membership function and fuzzy membership value had been taken to be the same, which led to the conclusion that the fuzzy sets do not follow the set theoretic axioms of exclusion and contradiction. For the complement of a normal fuzzy set, fuzzy membership function and fuzzy membership value are two different things, and the complement of a normal fuzzy set has to be defined accordingly. We shall further show how fuzzy randomness should be explained with reference to two laws of randomness defined for every fuzzy observation so as to make fuzzy statistical conclusions. Finally, we shall explain how randomness can be viewed as a special case of fuzziness defined in our perspective with reference to normal fuzzy numbers of the type [α, β, β]. Indeed every probability distribution function is a Dubois-Prade left reference function, and probability can be viewed in that way too.

18th International Conference of the North American Fuzzy Information Processing Society - NAFIPS (Cat. No.99TH8397)

Wiley series in probability and statistics, 2011

Messwerte kontinuierlicher physikalischer GroBen sind a priori unscharf und konnen mittels des Konzepts der unscharfen Zahlen und unscharfen Vektoren modelliert werden. Im Sinne einer quantitativen Verarbeitung solcher Daten ist es insbesondere notwendig, das klassische Konzept relativer Haufigkeiten fur reelle Stichproben auf sogenannte unscharfe relative Haufigkeiten fiir unscharfe Stichproben zu erweitern. Die unscharfe relative Haufigkeit einer Menge ist selbst eine unscharfe Zahl. Ausgehend von der Interpretation von Wahrscheinlichkeiten als Grenzwerte relativer Haufigkeiten ist es daher unumganglich, auch sogenannte unscharfe Wahrscheinlichkeitsverteilungen zu betrachten, fiir die die Wahrscheinlichkeit eines Ereignisses selbst eine unscharfe Zahl ist. Nach einer grundlegenden Abhandlung uber die wichtigsten algebraischen und topologischen Eigenschaften der Familie der unscharfen Zahlen und der Familie der d-dimensionalen unscharfen Vektoren ist daher der GroGteil der vorliegende Arbeit der Untersuchung zweier unterschiedlicher naturlicher Zugange zu unscharfen Wahrscheinlichkeitsverteilungen gewidmet: Jenem uber sogenannte unscharfe Wahrscheinlichkeitsdichten und einem speziellen Integrationsprozess ahnlich dem Aumann-Integral einerseits, und jenem uber die Verteilung unscharfer Zufallsvektoren andererseits. Es wird dabei insbesondere versucht, den hohen Grad an Gemeinsamkeit der durch die beiden Zugange induzierten unscharfen Wahrscheinlichkeitsverteilungen herauszustreichen. Daruberhinaus wird ein Starkes Gesetz der GroBen Zahlen fur unscharfe relative Haufigkeiten und unscharfe Wahrscheinlichkeitsverteilungen induziert von unscharfen Zufallsvektoren bezuglich verschiedener Metriken bewiesen und, in Verallgemeinerung reellwertiger stochastischer Prozesse, sogenannte unscharfe stochastische Prozesse definiert, und grundlegende Eigenschaften untersucht .

A number of similarities between fuzzy logic and probability can be identified. Historically, fuzzy logic is founded upon a probability based many valued logic of Lukasiewicz. The fuzzy membership function and the probability density function closely resemble as they both have similar shapes and have values that range between zero and one. Vague linguistic terms can be viewed as being probabilistic. The fuzzy rules of inference can be identified with probabilistic inference of inductive reasoning. The centroid method of calculating the output of fuzzy control is identical to that used in probability density functions, is namely the mean. Fuzzy logic could be probability as there as are so many close resemblances.

Cybernetics and Systems Analysis, 2020

An approach to finding the probabilistic characteristics of fuzzy events is proposed. The probabilities are described by triangular fuzzy numbers that can satisfy certain conditions. Examples of calculating fuzzy probabilities of fuzzy events are considered.

IAEME, 2019

This article deals with the relationship between Fuzzy Logic and the concept of Probability Function from discrete or continuous random variables. At first, basic concepts and properties are shown, both of functions corresponding to Fuzzy Logic, and functions of Probability Distribution and Probability Density. This includes some examples where properties of Fuzzy Logic and Probability functions are presented. Some applications of both concepts are also shown with illustrative exercises, reflecting, in turn, their relationship.

Austrian Journal of Statistics, 2018

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

Journal of Interdisciplinary Mathematics, 2018

This research starts from the concepts of conditional event and conditional probability introduced by de Finetti, Dubins and Scozzafava and from the concepts on fuzzy probability introduced by Zadeh, Klir, Yuan, and Yager. We propose a formal adjustment (but logically equivalent) to the definition of conditional event and related concepts, which allows us to give a definition of fuzzy event as an extension of a conditional event. Then, we suggest a formal modification (but logically equivalent) of the concept of conditional probability in the sense of de Finetti-Dubins, and we introduce some possible definitions of fuzzy event's probability as an extension of the conditional probability. Each of these definitions is an extension of the operational de Finetti definition of subjective conditional probability based on a coherent bet. Moreover, it contains a subjective evaluation, which is represented by a score function, that, in appropriate contexts, can be interpreted as a utility function.