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.Index Terms-fuzzy logic; mamdani method; machine learning; MCMC.

Marginal Likelihood Based Model Comparison in Fuzzy Bayesian Learning

“…Following on from the lines of [1] the inference problem is that of estimating θ for the underlying non-linear model function g(x; θ) given the data D which comprises of the set of input vectors X N = x i ∀i ∈ {1, 2, ..., N } and the set of outputs Y N = y i ∀i ∈ {1, 2, ..., N }. As introduced in [1], the function g(x; θ) is a fuzzy inference system given by the rule base Eq.…”

“…Consider a synthetic example similar to [1] where the downtime of an engineering operation needs to be predicted from two covariates -location risk and maintenance level. For the referential sets loc risk, maintenance, downtime the referential values are {LO, M ED, HI} , {P OOR, AV G, GOOD}, {LO, M ED, HI} respectively.…”

“…We use the same dataset and problem setting of modelling insurance risks for power plants as described in [1]. Please refer to [1] for detailed background of the problem and the rationale for applying the Fuzzy Bayesian methodology.…”

“…We use the same dataset and problem setting of modelling insurance risks for power plants as described in [1]. Please refer to [1] for detailed background of the problem and the rationale for applying the Fuzzy Bayesian methodology. The key idea is to encode the underwriters opinions as a fuzzy rule base expressing dependence between the three covariates -operation and maintenance (O&M), loss history, design & technology & fire risk (DnTnF) which are used in predicting the system uptime and consequently helping in making insurance premium calculations and underwriting decisions where the dataset is very small.…”

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In a recent paper [1] we introduced the Fuzzy Bayesian Learning (FBL) paradigm where expert opinions can be encoded in the form of fuzzy rule bases and the hyper-parameters of the fuzzy sets can be learned from data using a Bayesian approach. The present paper extends this work for selecting the most appropriate rule base among a set of competing alternatives, which best explains the data, by calculating the model evidence or marginal likelihood.

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