Physical Statistics or Statistical Physics? A Brief Overview of Bayesian Melding

For scientists, a model is typically a set of mathematical formulae that describe some natural phenomena of interest. One approach, often preferred by applied mathematicians, engineers, and physicists, is to discover physical laws of nature and express them as deterministic mathematical relationships among quantities that comprise such laws. At the opposite end of the spectrum, physical laws that gave rise to the data may be of little concern to students in an undergraduate statistics course, whose emphasis is on discovering empirical relationships among observable quantities and construct regression models that relate both observable and unobservable quantities, including random noise. Practising statisticians express known physical laws in their regression models; what makes a regression model empirical is that the behaviour of deviations (noise) from the mathematical formulae is an integral part of the model itself. For years, Bayesian hierarchical modelling has been the statistical framework of choice to integrate empirical and deterministic modelling. Bayesian melding is a more recent alternative statistical framework: it expresses physical laws as laws, thus without explicit noise terms, yet it still allows the focus to be placed on the behaviour of random quantities. In this overview, we discuss some philosophical underpinnings of the Bayesian melding approach, and through a toy example we illustrate the nuances of formulating a Bayesian melding model. We list some published and ongoing research in ecology, economics, engineering, epidemiology, and population dynamics that employ Bayesian melding; these examples suggest the potential for Bayesian melding to unify deterministic and statistical modelling approaches in general.