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

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