نمایندگی احتمالاتی و استنتاج تقریبی رویدادهای فازی نوع 2 در شبکه های بیزی با پارامترهای احتمال فاصله

It is necessary and challenging to represent the probabilities of fuzzy events and make inferences between them based on a Bayesian network. Motivated by such real applications, in this paper, we first define the interval probabilities of type-2 fuzzy events. Then, we define weak interval conditional probabilities and the corresponding probabilistic description. The expanded multiplication rule supporting interval probability reasoning. Accordingly, we propose the approach for learning the interval conditional probability parameters of a Bayesian network and the algorithm for its approximate inference. Experimental results show the feasibility of our method.

As a graphical representation of probabilistic causal relationships, Bayesian networks (BNs) are effective and widely used frameworks. A Bayesian network can be constructed by means of statistical learning from sample data. Probabilistic inferences can be done by computing products of conditional probabilities from Bayesian networks (Pearl, 1988). In real-world applications, a Bayesian network is developed to describe the causal relationships of exact sample data, while it is often desirable to determine the causality between fuzzy random events based on the Bayesian network. For example, we want to know the probability of a fuzzy event, such as “a low atmospheric pressure causes a heavy rain”. For this subject, we need to discuss the following problems: • How to represent the probability of a fuzzy event? • How to make inferences between fuzzy events in a Bayesian network? Actually, the traditional probability representation of fuzzy event A is View the MathML sourceP(A)=∑xi∈ΩμA(xi)P(xi), Turn MathJax on where ΩΩ is the sample space and μAμA is the membership function. Note that P(A)P(A) is a crisp value. It is hard to image that the uncertainty concept “heavy rain” has a crisp probability, and the reason is that we describe a linguistic variable by a certain membership function. A linguistic variable may mean different things to different people, and thus we always represent the uncertainties by a range or interval of values. Fortunately, Zadeh (1975) introduces type-2 fuzzy sets to convey the uncertainties in membership functions of ordinary fuzzy sets. As for the probabilistic representation and inferences of random variables, in a general Bayesian network, the causal relationships among them are represented by crisp probabilities. Clearly, it is infeasible to represent and infer the probabilistic causal relationships of type-2 fuzzy events directly by the general Bayesian network. In this paper, aiming at the problems pointed out above, we focus on the representation and inference of type-2 fuzzy events in a Bayesian network. It is known that the methods of representation and inferences of imprecise probabilities are well applied to describing the uncertainty knowledge (Walley, 1991 and Cano and Moral, 2002). In recent years, credal networks, as a kind of qualitative abstraction of general Bayesian networks, have been adopted as the standard for graphical models that extend Bayesian networks to deal with imprecision with probabilities (Cozman, 2000, Cozman, 2005 and Cano et al., 2007). However, these two types of methods adopt probability intervals to represent the uncertainties of random variables. Therefore, the computation of probability intervals themselves cannot be made under a certain theoretical basis, compared to the probability computation based on the probability theory. The propagation of probability intervals during inferences cannot be guaranteed to be sound theoretically as well. The probability theory is the foundation of Bayesian networks, and the probabilistic inferences are fulfilled based on the principles of the conditional probability, multiplication rule, Bayes formula, etc. To make the representation and inference of type-2 fuzzy events with a Bayesian network, it is necessary to bridge the gap between crisp probabilities and the uncertain concept. Thus we will have to extend the traditional probability theory by incorporating the characteristics of type-2 fuzzy sets. Fortunately, interval probability theory has been accepted as a formal method to representing uncertainties in an imprecise manner by typical interval values (Weichselberger, 2000 and Weichselberger et al., 2003). Interval probabilities are applied to describe the semantics of imprecise probabilities and uncertain knowledge (Gilbert et al., 2003 and Tanaka et al., 2004). Therefore, in this paper we will adopt the interval probability theory, instead of probability intervals, as the backbone of representation and inferences of type-2 fuzzy events with a Bayesian network. To make the interval probability theory be suitable for the representation and inferences of type-2 fuzzy events, we first define interval probabilities, interval conditional probabilities and weak interval conditional probabilities of fuzzy events. By means of the Bayesian network with interval probabilistic parameters, the causal relationships among fuzzy events can be represented. Further, aiming at the feasible inferences between fuzzy events in a Bayesian network, we define bound-limited weak interval conditional probabilities and expand the corresponding multiplication rule for joint probability distribution. Based on the definitions of weak interval conditional probabilities, we mainly give a method for learning interval conditional probability tables of fuzzy random events in a Bayesian network. Then, we give a Gibbs sampling algorithm for inferences of the Bayesian network with interval probabilities. Preliminary experiments show that our method is feasible. The remainder of this paper is organized as follows: Section 2 introduces related work. Section 3 first defines the interval probabilities, interval conditional probabilities of fuzzy events. Then, weak interval conditional probabilities, the corresponding probabilistic description, and the expanded multiplication rule are defined. Section 4 presents the method for learning interval conditional probability tables in a Bayesian network. Section 5 gives the algorithm for approximate inference in Bayesian networks with interval probability parameters. Section 6 shows the experimental results. Section 7 concludes and discusses the future work.

Aiming at the representation and inference of the causalities between fuzzy random events based on a Bayesian network, in this paper, we are to bridge the gap between the uncertainty of fuzzy events and the crisp probabilities in a Bayesian network. We adopt the interval probability theory as the backbone foundation to extend the general Bayesian network. First, based on type-2 fuzzy sets, we define the interval probability, interval conditional probability and weak interval conditional probability of fuzzy random variables. As well, we also give the corresponding probabilistic description. Then, we give the method of obtaining the interval probability parameters of fuzzy events based on the definition of the bound-limited weak interval conditional probability. Accordingly, we give an approximate inference algorithm of Bayesian networks with interval probability parameters. Experimental result shows that out proposed methods are feasible. To further explore the method for obtaining more closer lower and upper probabilities is our current focus. Meanwhile, the approaches proposed in this paper also raise some other interesting research issues. Based on our proposed approach in this paper, the probability-network-based scene prediction and decision can be made. Probabilistic causal knowledge implied in sample data with fuzzy events can be discovered accordingly. As well, the Bayesian network with interval probability parameters can be constructed from the sample data with interval values. These research issues are right our future work.