The Metropolis-Hastings algorithm, a handy tool for the practice of environmental model estimation: illustration with biochemical oxygen demand data

Environmental scientists often face situations where: (i) stimulus-response relationships are non-linear; (ii) data are rare or imprecise; (iii) facts are uncertain and stimulus-responses relationships are questionable. In this paper, we focus on the first two points. A powerful and easy-to-use statistical method, the Metropolis-Hastings algorithm, allows the quantification of the uncertainty attached to any model response. This stochastic simulation technique is able to reproduce the statistical joint distribution of the whole parameter set of any model. The Metropolis-Hastings algorithm is described and illustrated on a typical environmental model: the biochemical oxygen demand (BOD). The aim is to provide a helpful guideline for further, and ultimately more complex, models. As a first illustration, the MH-method is also applied to a simple regression example to demonstrate to the practitioner the ability of the algorithm to produce valid results