Multilevel Bayesian Parameter Estimation in the Presence of Model Inadequacy and Data Uncertainty

TL;DR

This paper introduces a hierarchical Bayesian framework to address model inadequacy and data uncertainty, enhancing inference and prediction accuracy in scientific applications.

Contribution

It develops a comprehensive Bayesian approach that explicitly accounts for model bias and measurement noise, improving the reliability of scientific inference.

Findings

Framework effectively quantifies model inadequacy and data uncertainty.

Applicable to diverse scientific, engineering, and statistical problems.

Clarifies the relationship between different types of uncertainty.

Abstract

Model inadequacy and measurement uncertainty are two of the most confounding aspects of inference and prediction in quantitative sciences. The process of scientific inference (the inverse problem) and prediction (the forward problem) involve multiple steps of data analysis, hypothesis formation, model construction, parameter estimation, model validation, and finally, the prediction of the quantity of interest. This article seeks to clarify the concepts of model inadequacy and bias, measurement uncertainty, and the two traditional classes of uncertainty: aleatoric versus epistemic, as well as their relationships with each other in the process of scientific inference. Starting from basic principles of probability, we build and explain a hierarchical Bayesian framework to quantitatively deal with model inadequacy and noise in data. The methodology can be readily applied to many common inference and prediction problems in science, engineering, and statistics.