Fuzzy Bayesian Learning

Abstract--In this paper we propose a novel approach for learning from data using rule based fuzzy inference systems where the model parameters are estimated using Bayesian inference and Markov Chain Monte Carlo (MCMC) techniques. We show the applicability of the method for regression and classification tasks using synthetic data-sets and also a real world example in the financial services industry. Then we demonstrate how the method can be extended for knowledge extraction to select the individual rules in a Bayesian way which best explains the given data. Finally we discuss the advantages and pitfalls of using this method over state-of-the-art techniques and highlight the specific class of problems where this would be useful. ROBABILITY theory and fuzzy logic have been shown to be complementary [1] and various works have looked at the symbiotic integration of these two paradigms [2], [3] including the recently introduced concept of Z-numbers [4]. Historically fuzzy logic has been applied to problems involving imprecision in linguistic variables, while probability theory has been used for quantifying uncertainty in a wide range of disciplines. V arious generalisations and extensions of fuzzy sets have been proposed to incorporate uncertainty and vagueness which arise from multiple sources. For example, the type-2 fuzzy [5], [6] sets and type-n fuzzy sets [5] can include uncertainty while defining the membership functions themselves. Intuitionistic fuzzy sets [7] additionally introduce the degree of non-membership of an element to take into account that there might be some hesitation degree and the degree of membership and non-membership of an element might not always add to one. Non-stationary fuzzy sets [8] can model variation of opinion over time by defining a collection of type 1 fuzzy sets and an explicit relationship between them. Fuzzy multi-sets [9] generalise crisp sets where multiple occurrences of an element are permitted. Hesitant fuzzy sets [10] have been proposed from the motivation that the problem of assigning a degree of membership to an element is not because of a margin of error (like Atanassov's intuitionistic fuzzy sets) or a possibility distribution on possibility values (e.g. Formally these can be viewed as fuzzy multi-sets but with a different interpretation.