Bayesian Approach for Inconsistent Information

Authors

M. Stein, National University of Singapore
Michael Beer, University of LiverpoolFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

1-2013

Comments

Technical Report: UTEP-CS-12-26c

To appear in Information Sciences.

Abstract

In engineering situations, we usually have a large amount of prior knowledge that needs to be taken into account when processing data. Traditionally, the Bayesian approach is used to process data in the presence of prior knowledge. Sometimes, when we apply the traditional Bayesian techniques to engineering data, we get inconsistencies between the data and prior knowledge. These inconsistencies are usually caused by the fact that in the traditional approach, we assume that we know the {\it exact} sample values, that the prior distribution is {\it exactly} known, etc. In reality, the data is imprecise due to measurement errors, the prior knowledge is only approximately known, etc. So, a natural way to deal with the seemingly inconsistent information is to take this imprecision into account in the Bayesian approach -- e.g., by using fuzzy techniques. In this paper, we describe several possible scenarios for fuzzifying the Bayesian approach. Particular attention is paid to the interaction between estimated imprecise parameters.

In this paper, to implement the corresponding fuzzy versions of the Bayesian formulas, we use straightforward computations of the related expression -- which makes our computations reasonably time-consuming. Computations in the traditional (non-fuzzy) Bayesian approach are much faster -- because they use algorithmically efficient reformulations of the Bayesian formulas. We expect that similar reformulations of the fuzzy Bayesian formulas will also drastically decrease the computation time and thus, enhance the practical use of the proposed methods.