Learning parameters of fuzzy Bayesian Network based on imprecise observations

In recent years, Bayesian Network has become an important modeling method for decision making problems of real-world applications. In this paper learning parameters of a fuzzy Bayesian Network (BN) based on imprecise/fuzzy observations is considered, where imprecise observations particularly refers to triangular fuzzy numbers. To achieve this, an extension to fuzzy probability theory based on imprecise observations is proposed which employs both the "truth" concept of Yager and the Extension Principle in fuzzy set theory. In addition, some examples are given to demonstrate the concepts of the proposed idea. The aim of our suggestion is to be able to estimate joint fuzzy probability and the conditional probability tables (CPTs) of Bayesian Network based on imprecise observations. Two real-world datasets, Car Evaluation Database (CED) and Extending Credibility (EC), are employed where some of attributes have crisp (exact) and some of them have fuzzy observations. Estimated parameters of the CED's corresponding network, using our extension, are shown in tables. Then, using Kullback-Leibler divergence, two scenarios are considered to show that fuzzy parameters preserve more knowledge than that of crisp parameters. This phenomenon is also true in cases where there are a small number of observations. Finally, to examine a network with fuzzy parameters versus the network with crisp parameters, accuracy result of predictions is provided which shows improvements in the predictions.