On the assimilation of uncertain physical knowledge bases: Bayesian and non-Bayesian techniques

Abstract

We study the assimilation of uncertain physical knowledge across space and time, and we discuss the resulting issues of environmental inference and prediction. The term “assimilation” refers to a general framework that includes physical knowledge integration and processing activities in a variety of scientific disciplines (atmospheric data analysis and weather forecasting, environmental pollution mapping, exposure assessment, etc.). The choice of an adequate assimilation approach is primarily a conceptual modelling affair that is supported by the physical and logical features of the environmental situation under consideration, rather than merely relying on pure inductive schemes and statistical arguments. In many cases, the assimilation approach involves conditionalization techniques which enable it to merge various forms of site-specific data with modelling frameworks. We distinguish between interpretive conditionalization (which refers to the physical and epistemic characteristics of assimilation modelling) and formal conditionalization (which is a purely mathematical construction), and emphasize the importance of an adequate choice of the latter to represent the former. We point out the considerable improvements of the operational Bayesian conditionalization approach over the standard Bayesian rule. Statistical learning techniques are useful at the descriptive/correlation level of scientific inquiry, whereas the level of explanation/prediction is the domain of advanced operational techniques. An operational approach is based on an epistemic view in which the relevant natural processes are represented in terms of spatiotemporal random fields and the probability laws are expressed in terms of physically meaningful operators. The spirit of this view is to keep the field solutions consistent with the site-specific information (hard data, uncertain observations, etc.) while exactly satisfying the constraints arising from general knowledge sources (physical laws, primitive equations, scientific theories, etc.). Natural field probabilities in space and time are derived for non-Bayesian operational conditionals based on deductively sound inference, which are valid for a wide variety of knowledge bases. Analytical and numerical comparisons are made between Bayesian and non-Bayesian conditionals, and insight is gained in terms of examples and applications which cut across various earth science disciplines. Operational Bayesian conditionalization is a powerful and versatile component of assimilation modelling with many applications in environmental sciences, although in certain cases a non-Bayesian conditional (based on deductive logic and the characterization of physical connection) may provide a meaningful description of the data assimilation framework suggested by the laws of nature.