Multilevel Bayesian Analysis of Biophysical Data in the Presence of Model Inadequacy and Measurement Error

Model inadequacy and measurement uncertainty are two of the most confounding aspects of inference and prediction in quantitative sciences. The process of scientific inference and prediction involve multiple steps of data analysis, hypothesis formation, model construction, parameter estimation, model validation, and finally prediction of the quantity of interest. This work seeks to clarify the concepts of model inadequacy, model bias, and measurement uncertainty, along with the two traditional classes uncertainty: aleatoric vs. epistemic, as well as their relashionships with each other. Starting from basic principles of probability, we build and explain a hierarchical Bayesian framework to quantitatively deal with model inadequacy and noise in data. We explain how this general approach can resolve many existing logical paradoxes that frequently appear in biophysical datasets. As an illustrative problem, we apply the methodology to an in-vitro dataset of the growth of C3A liver tumor cells subject to significant imaging background noise. We then explain how this general approach can retrieve the unknown quantities of interest from the available noisy data without any logical inconsistencies, such as obtaining negative values for quantities that are known to be inherently positive.