Vagueness and Bayesian probability

This paper is a response to Michael Laviolette and John W. Seaman Jr.'s ( ibid. vol.2, no.1, p.4 (1994)) position paper “The efficacy of fuzzy representations of uncertainty,” which criticizes fuzzy representations of uncertainty, and suggests that Bayesian probability can do better. The commenter argues that the author's make some misleading comments about Bayesian probability, and he briefly discusses the problem of giving a satisfactory interpretation of membership functions     

Comments on “The efficacy of fuzzy representations ofuncertainty”

The author expresses agreement with the original paper, by M. Laviolette and J.W. Seaman (ibid., p.4-15), saying that probability is the only satisfactory measure of one's personal uncertainty about the world. The reason for this emphatic statement is that there exist several axiom systems that produce, as theorems, the rules of probability. In particular, the author criticises the use of fuzzy logic as an alternative     

On the alleged superiority of probabilistic representation ofuncertainty

The author identifies some points of his agreement or disagreement with the position taken by Laviolette and Seaman (1994) in their debate paper. In particular, the author argues that some statements regarding the superiority of probabilistic representation of uncertainty, which are made by Laviolette and Seaman in their paper, are untenable     

A random set approach to the optimization of uncertain structures

The single-objective optimization of structures, whose parameters are assigned as fuzzy numbers or fuzzy relations, is presented in this paper as a particular case of the random set theory and evidence theory approach to uncertainty. Some basic concepts concerning these theories are reviewed and the relationships among interval analysis, convex modeling, possibility theory and probability theory are pointed out. In this context a frequentistic view of fuzzy sets makes sense and it is possible to calculate bounds on the probability that the solution satisfies the constraints. Some special but useful cases illustrate in detail the meaning of the approach proposed and its links with a recent formulation conceived within the context of convex modeling. Some theorems allow a very efficient computational procedure to be set up in many real design situations. Two numerical examples illustrate the model presented.     

证据推理的进展及存在的问题

综合论述了证据推理的进展和存在的问题,把证据推理与贝叶斯推理、模糊逻辑推理、基于规则的推理、随机集理流体进行了横向比较,讨论了其间的必然的联系和为语气推理的发展的基础,随后总结了证据推理的改进及其在概率范围在模糊集等方面推广的进展,最后提出了证据推理的广泛应用及其研究方向。     

正态模糊集合——Fuzzy集理论的新拓展

直觉模糊集（intuitionistic fuzzy sets）、区间值模糊集（interval-valued fuzzy sets）以及Vague集对普通fuzzy集的扩展是给出了隶属度的上下限,把隶属度从[0,1]区间中的一个单值推广到了[0,1]的子区间。但是该子区间犹如一个黑洞,隶属度在其内部的分布情况我们无从知晓,即这个子区间中的每一个值是等可能地作为元素的隶属度还是区间中的某些值较另外的值有更大的可能性呢？为了清晰的刻画出元素的隶属度在[0,1]区间中的分布情况,本文通过对投票模型的分析及正态分布理论,提出了一种新的模糊集合——正态模糊集合,同时对正态模糊集合的交、并、补等基本运算性质进行了讨论,文章最后对正态模糊集与fuzzy集、直觉模糊集的相互关系也作出了详细阐述。正态模糊集合是模糊集合理论的进一步推广,为我们处理模糊信息提供了一种全新的思想方法。     

Possibility theory and statistical reasoning

Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev's) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out.     

Generalized probabilities of fuzzy events from fuzzy belief structures

We extend Shafer's theory of evidence to include the ability to have belief structures involving fuzzy sets. We then obtain under the condition of Bayesian belief structure a whole family of possible definitions for the probability of fuzzy sets. We also suggest a procedure for including belief qualification in pruf.     

Some new results concerning random sets and fuzzy sets

This paper is a continuation of past work showing direct connections between fuzzy set theory and classical probability theory through the use of random sets. This includes characterizations of random intervals one-point-coverage equivalent to fuzzy sets, determination of all nested random sets equivalent to fuzzy sets, the solution of the one- and multiple-point coverage problems for random sets in finite spaces, and the use of entropy for ordering random sets within the class of one-point-coverage-equivalent ones for any given fuzzy set. Finally, a connection between random variables, random sets, and fuzzy sets is pointed out.     

Conversion of probabilistic information into fuzzy sets for engineering decision analysis

To carry out seismic hazard analysis in the framework of fuzzy set theory, it may become necessary to convert probabilistic information regarding some of the variables into triangular or trapezoidal fuzzy sets. In this paper, three approaches for converting probabilistic information, represented by a probability distribution, into an equivalent triangular or trapezoidal fuzzy set are discussed. In all the three approaches, the probability distribution is first converted into a probabilistic fuzzy set, which is then converted into the equivalent triangular or trapezoidal fuzzy set. The first approach is based on the method of least-square curve fitting, the second approach is based on the conservation of uncertainty (represented by the entropy) associated with the probabilistic fuzzy set in a mean square sense, and the third approach is based on the minimisation of Hausdorff distance (HD) between the probabilistic and the equivalent fuzzy sets. The effectiveness of these approaches in preserving the entropy as well as in preserving the elements of the fuzzy set and their corresponding grades of membership are also discussed with the help of a numerical example of obtaining equivalent fuzzy set for peak ground acceleration. It is found that the approach based on minimisation of Hausdorff distance provides a simple and efficient way for converting the probabilistic information into an equivalent fuzzy set.     

Advocating the Use of Imprecisely Observed Data in Genetic Fuzzy Systems

In our opinion, and in accordance with current literature, the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet. In particular, we question the existence of a set of problems for which the use of fuzzy rules, in combination with genetic algorithms, produces more robust models, or classifiers that are inherently better than those arising from the Bayesian point of view. We will show that this set of problems actually exists, and comprises interval and fuzzy valued datasets, but it is not being exploited. Current genetic fuzzy classifiers deal with crisp classification problems, where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions, with a convenient linguistic interpretation. Provided that the customary use of fuzzy sets in statistics is vague data, we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems. The same can be said about genetic fuzzy models: the use of a scalar fitness function assumes crisp data, where fuzzy models, a priori, do not have advantages over statistical regression.     

The efficacy of fuzzy representations of uncertainty

Advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments. These are: 1) the reality hypothesis, which holds that imprecision is an inherent property of the world external to an observer; 2) the subjectivity hypothesis, which holds that probability is an exclusively objective measure of uncertainty, and that therefore subjective uncertainty can only be represented with fuzzy sets; 3) the behaviorist hypothesis, which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty; 4) the “probability as fiction” hypothesis, which claims that probability does not comprise a field of study in its own right; and 5) the superset hypothesis, which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment. We discuss and criticize all five. We then criticize the argument that fuzziness represents a type of uncertainty distinct from probability, and also the inordinate complexity of fuzzy methods. We present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples     

基于模糊集和DS证据理论的信息安全风险评估方法*

在信息安全风险评估过程中,存在着很多不确定和模糊的因素,针对专家评价意见的不确定性和主观性问题,提出了一种将模糊集理论与DS证据理论进行结合的的风险评估方法。首先,根据信息安全风险评估的流程和要素,建立风险评估指标体系,确定风险影响因素;其次,通过高斯隶属度函数,求出专家对各影响因素的评价意见隶属于各个不同评价等级的程度;再次,将其作为DS理论所需的基本概率分配,引入基于矩阵分析和权值分配的融合算法综合多位专家的评价意见;最后,结合贝叶斯网络模型的推理算法,得出被测信息系统所面临的风险大小,并对其进行分析。结果显示,将模糊集理论和DS证据理论应用到传统贝叶斯网络风险评估的方法,在一定程度上能够提高评估结果的客观性。     

Fuzzy rough set on probabilistic approximation space over two universes and its application to emergency decision‐making

Probabilistic approaches to rough sets are still an important issue in rough set theory. Although many studies have been written on this topic, they focus on approximating a crisp concept in the universe of discourse, with less effort on approximating a fuzzy concept in the universe of discourse. This article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space over two universes. We first present the definition of a lower and upper approximation of a fuzzy set with respect to a probabilistic approximation space over two universes by defining the conditional probability of a fuzzy event. That is, we define the rough fuzzy set on a probabilistic approximation space over two universes. We then define the fuzzy probabilistic approximation over two universes by introducing a probability measure to the approximation space over two universes. Then, we establish the fuzzy rough set model on the probabilistic approximation space over two universes. Meanwhile, we study some properties of both rough fuzzy sets and fuzzy rough sets on the probabilistic approximation space over two universes. Also, we compare the proposed model with the existing models to show the superiority of the model given in this paper. Furthermore, we apply the fuzzy rough set on the probabilistic approximation over two universes to emergency decision‐making in unconventional emergency management. We establish an approach to online emergency decision‐making by using the fuzzy rough set model on the probabilistic approximation over two universes. Finally, we apply our approach to a numerical example of emergency decision‐making in order to illustrate the validity of the proposed method.     

An extension of statistical decision theory with information theoretic cost functions to decision fusion: Part I

A review of information theory and statistical decision theory has led to the recognition that decisions in statistical decision theory can be interpreted as being determined by the similarity between the distribution of probabilities obtained from measurements and characteristic distributions of probabilities representing the members of the set of decisions. The obscure interpretation was found during a review of statistical decision theory for the special case where the cost function of statistical decision theory is an information theoretic cost function.Additional research has found that the resulting information theoretic decision rule has a number of interesting characteristics that may have previously been recognized in terms of mathematical interest, but until now have not been recognized for their implications for information fusion. Bayesian probability theory has been criticized for problematic changes in decisions when hypotheses and decisions are reorganized to different levels of abstraction, weak justification for the selection of prior probabilities, and the need for all probability density functions to be defined. The characteristics of the information theoretic decision rule show that the decisions are less sensitive to changes in the reorganization of hypothesis and decision sets to different levels of abstraction in comparison to Bayesian probability theory. Extension of the information theoretic rule to a fusion rule (to be provided in a companion paper) will be shown to provide increased justification for the selection of prior probabilities through the adoption of Laplace’s principle of indifference. The criticism of the need for all probability density functions can be partially mitigated by arguing that the hypothesis abstraction levels can be selected so that all the probability density functions may be obtained. Further refutation of the third criticism will require that the assumption that the probability density functions are not definitively known but may be ambiguous as well and will not be pursued as a line of inquiry within the two companion papers.     

基于贝叶斯最优分类器的多源模糊信息融合方法

为了使传统的贝叶斯最优分类器能够处理模糊信息和实现推理过程的自动化, 在这篇文章里我们将模糊信息嵌入到贝叶斯最优分类器中, 形成新的贝叶斯最优分类器. 它不但能有效地处理模糊信息, 而且还保留了贝叶斯最优分类器的学习性能. 再者, 根据模糊集理论的发展, vague 集也嵌入到贝叶斯最优分类器中形成 vague 贝叶斯最优分类器.它能同时模拟模糊信息正、反两方面的特征. 进一步, 提出能同时处理正、反和不确定三方面模糊信息的集对贝叶斯最优分类器. 最终, 为了实现贝叶斯最优分类器的自动推理, 提出一种基于知识的人工神经网络 (KBANN) 的贝叶斯最优分类器. 它不仅降低了贝叶斯最优分类器的计算量, 而且还改善了它的分类学习质量.     

考虑模糊集合相关性的T范式和T协范式算子

在仔细分析模糊逻辑和模糊集合论中最常用的Zadeh算子的基础上，研究了模糊集合相关性对T范式和T协范式算子运算结果的影响，并分析了Zadeh算子之所以不满足矛盾律和排中律的根本原因．最终提出了考虑集合相关性的新的T范式和T协范式算子，而且证明了该算子在不违反模糊集合的模糊性的基础上能够使模糊集合的运算满足排中律和矛盾律，并能使模糊集合理论和经典集合理论很好地统一．     

FUZZINESS VS. PROBABILITY

Fuzziness is explored as an alternative to randomness for describing uncertainty. The new sets-as-points geometric view of fuzzy sets is developed. This view identifies a fuzzy set with a point in a unit hypercube and a nonfuzzy set with a vertex of the cube. Paradoxes of two-valued logic and set theory, such as Russell's paradox, correspond to the midpoint of the fuzzy cube. The fundamental questions of fuzzy theory—How fuzzy is a fuzzy set? How much is one fuzzy set a subset of another?—are answered geometrically with the Fuzzy Entropy Theorem, the Fuzzy Subsethood Theorem, and the Entropy-Subsethood Theorem. A new geometric proof of the Subsethood Theorem is given, a corollary of which is that the apparently probabilistic relative frequency n_A /N turns out to be the deterministic subsethood S(X, A), the degree to which the sample space X is contained in its subset A. So the frequency of successful trials is viewed as the degree to which all trials are successful. Recent Bayesian polemics against fuzzy theory are examined in light of the new sets-as-points theorems.     

Application of fuzzy logic and variable precision rough set approach in a remote monitoring manufacturing process for diagnosis rule induction

Rough set has been shown to be a valuable approach to mine rules from a remote monitoring manufacturing process. In this research, an application of the fuzzy set theory with the fuzzy variable precision rough set approach for mining the causal relationship rules from the database of a remote monitoring manufacturing process is presented. The membership function in the fuzzy set theory is used to transfer the data entries into fuzzy sets, and the fuzzy variable precision rough set approach is applied to extract rules from the fuzzy sets. It is found that the induced rules are identical to the practical knowledge and fault diagnosis thinking of human operators. The induced rules are then compared with the rules induced by the original rough set approach. The comparison shows that the rules induced by the fuzzy rough set are expressed in linguistic forms, and are evaluated by plausibility and future effectiveness measures. The fuzzy rough set approach, being less sensitive to noisy data, induces better rules than the original rough set approach.