An Interval Type-2 Version of Bayes Theorem Derived from Interval Probability Range Estimates Provided by Subject Matter Experts
This work addresses the challenge of handling uncertainty in Bayesian inference for real-world applications where precise data is unavailable, though it is incremental as it builds on existing interval and fuzzy logic methods.
The paper tackles the problem of applying Bayesian inference with imprecise input probabilities, often provided as interval estimates by subject matter experts, by developing an interval type-2 version of Bayes Theorem and a novel algorithm for encoding these intervals into fuzzy membership functions. The result is a method that avoids inconsistencies and generalizes previous work on interval encoding.
Bayesian inference is widely used in many different fields to test hypotheses against observations. In most such applications, an assumption is made of precise input values to produce a precise output value. However, this is unrealistic for real-world applications. Often the best available information from subject matter experts (SMEs) in a given field is interval range estimates of the input probabilities involved in Bayes Theorem. This paper provides two key contributions to extend Bayes Theorem to an interval type-2 (IT2) version. First, we develop an IT2 version of Bayes Theorem that uses a novel and conservative method to avoid potential inconsistencies in the input IT2 MFs that otherwise might produce invalid output results. We then describe a novel and flexible algorithm for encoding SME-provided intervals into IT2 fuzzy membership functions (MFs), which we can use to specify the input probabilities in Bayes Theorem. Our algorithm generalizes and extends previous work on this problem that primarily addressed the encoding of intervals into word MFs for Computing with Words applications.