Application of the transferable belief model to diagnostic problems

Abstract

Short presentation of the most relevant elements of the transferable belief model and its use for problems related to the diagnostic process. These examples illustrate the use of the transferable belief model and in particular of the Generalized Bayesian Theorem. Uncertainty is classically represented by probability functions, and diagnostic in an environment poised by uncertainty is usually handled through the application of the Bayesian theorem that permits the computation of the posterior probability over the diagnostic categories given the observed data from the prior probability over the same categories. We show here that the whole problem admits a similar solution when uncertainty is quantified by belief functions as in the transferable belief model. The classical Bayesian theorem admits a generalization within the transferable belief model (TBM) that we called the Generalized Bayesian Theorem (Smets, 1978, 1981, 1993a). This theorem seems to have been often overlooked, and the use of conditional belief functions for diagnostic problems neglected. The Generalized Bayesian Theorem (GBT) permits the computation of the conditional belief over the diagnostic classes given an observed data from the knowledge of the set of the conditional beliefs about which data will be observed when the case belongs to a given diagnostic category. Loosely expressed, this inversion theorem permits to pass from a belief on the symptoms given the diseases to a belief on the diseases given the symptoms. We present hereafter four examples of diagnostic process within the TBM, and compared the TBM solution with its obvious contender, the probability solution. The examples are analyzed in detail in order to give a clear understanding of the exact use of the TBM and its GBT. We restrict ourself to ‘simple’ examples, cases of complex systems and common or dependent causes are not tackled. Our aim is in showing how the classical Bayesian theorem can be extended and applied within the TBM framework.