Bayesian Inference Based on Interval-Valued Prior Distributions and Likelihoods

Although Bayesian inference has been successful in many applications, its serious limitation is the requirement that exact prior probabilities be available. It has increasingly been recognized that this requirement is often not realistic. To overcome this limitation of classical Bayesian inference, we investigate a generalized Bayesian inference, in which prior probabilities as well as likelihoods are interval-valued. Employing the tools of interval analysis and the theory of imprecise probabilities, we develop a method for exact calculation of interval-valued posterior probabilities for given interval-valued prior probabilities and precise or interval-valued likelihoods. This method is further generalized for fuzzy likelihood and fuzzy probabilities later. The classical Bayesian inference is a special case of our method.