An Interval Type-2 Version of Bayes Theorem Derived from Interval Probability Range Estimates Provided by Subject Matter Experts

ArXiv ID: 2509.08834 “View on arXiv”

Authors: John T. Rickard, William A. Dembski, James Rickards

Abstract

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.

Keywords: Bayesian Inference, Interval Type-2 Fuzzy Logic, Uncertainty Quantification, Decision Making, Risk Analysis, Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents dense mathematical derivations involving interval type-2 fuzzy sets and interval arithmetic, but lacks any backtesting, statistical validation, or real-world data implementation, focusing instead on theoretical algorithm construction.

  flowchart TD
    A["Research Goal<br>Develop an Interval Type-2<br>Bayes Theorem using SME Intervals"] --> B{"Key Methodology"}
    B --> C["Step 1: Encode SME Intervals<br>into IT2 Fuzzy MFs"]
    B --> D["Step 2: Apply Conservative<br>IT2 Bayes Theorem"]
    
    C --> E["Computational Process<br>Integration of IT2 MFs<br>via Interval Probability"]
    D --> E
    
    E --> F["Key Findings & Outcomes"]
    F --> G["Valid IT2 Bayesian Inference<br>Prevents Input Inconsistencies"]
    F --> H["Flexible Algorithm for<br>SME Interval Encoding"]