Fuzzy Bayesian inference

Bayesian methods provide formalism for reasoning about partial beliefs under conditions of uncertainty. Given a set of exhaustive and mutually exclusive hypotheses, one can compute the probability of a hypothesis for given evidence using the Bayesian inversion formula. In the Bayesian inference, the evidence could be a single atomic proposition or multi-valued. For multi-valued evidence, these values could be discrete, continuous, or fuzzy. For continuous-valued evidence, the density functions used in the Bayesian inference are difficult to be determined in many practical situations. Complicated laboratory testing and advance statistical techniques are required to estimate the parameters of the assumed type of distribution. Using the proposed fuzzy Bayesian approach, formulation is derived to estimate the density function from the conditional probabilities of the fuzzy-supported values. It avoids the complicated testing and analysis, and it does not require the assumption of a particular type of distribution. The estimated density function in our approach is proved to conform to two axioms in the theorem of the probability. Example is provided in the paper.