  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.
