Bayesian Inference

The preceding chapter discussed how scientific research is always carried out under conditions of incomplete information. We never know ‘the whole story’, much remains uncertain. Scientists therefore increasingly turn to probability theory when applying their models (e.g. Hartig et al., J Biogeogr 39:2240–2252, 2012; Jaynes, Probability theory: The logic of science. Cambridge University Press, 2003; Ogle and Barber, Bayesian Data—Model Integration in Plant Physiological and Ecosystem Ecology. In U Lüttge, W Beyschlag, J Murata (Eds.), Progress in Botany, Progress in Botany (pp. 281–311). Springer, 2008; Sivia and Skilling, Data Analysis: A Bayesian Tutorial (2nd ed.). Oxford University Press, 2006; Van Oijen & Brewer, Probabilistic Risk Analysis and Bayesian Decision Theory, SpringerBriefs in Statistics. Springer International Publishing, 2022). This is the approach that we take in this book too. We show how defining all our uncertainties as probability distributions allows for rigorous reduction of those uncertainties when new data come in. The approach that we are presenting is known in the literature under many different names, including Bayesian calibration, data assimilation, model-data fusion and inverse modelling. Whilst the different names refer to different applications of modelling, they all share the idea of specifying probability distributions and modifying them according to the rules of probability theory (in particular as we shall see, Bayes’ Theorem) when new data become available. This manner of drawing probabilistic conclusions is called Bayesian inference. But how does it work exactly?