A Unified Framework for Quantifying Uncertainty: Synthesizing Classical and Bayesian Probabilistic Inference

Abstract

The quantification of uncertainty in scientific modeling is fundamentally divided between the classical (frequentist) and Bayesian paradigms, compelling practitioners to adopt an either/or approach that often discards valuable information. This paper introduces a novel, unified mathematical framework that synthesizes the inferential outputs of both paradigms. Leveraging the likelihood function as a common foundation, the framework employs a linear pooling operator to combine the classical confidence distribution and the Bayesian posterior distribution into a single, more comprehensive representation of uncertainty, the primary output is a "Unified Uncertainty Interval" (UUI), which inherits both the long-run frequency guarantees of confidence intervals and the intuitive, belief-based interpretation of credible intervals. Case studies involving binomial proportion estimation, particularly under conditions of prior-data conflict, demonstrate that the UUI provides a robust and balanced measure of uncertainty, the framework offers a pragmatic solution to bridge the gap between classical and Bayesian approaches, providing a richer, more nuanced tool for decision-making under uncertainty and moving beyond paradigmatic dogmatism towards a more holistic inferential practice.