What is a prior and how to derive one?

Application of Bayesian statistics requires eliciting prior distributions, an important first step that is often ignored. The difficulty in prior elicitation is largely due to the vague definition of the prior. Furthermore, formal methods for deriving priors are mostly focused on deriving priors with least amount of information (e.g., the reference prior). In practice, we often resort to a class of “non-informative” or “vague” priors when using relatively simple models. These priors are usually informative in some way and can lead to unintended consequences. In this presentation, I discuss the meaning of a prior distribution from an empirical Bayes perspective, which is the “centre of gravity” of similar (exchangeable) units. Based on this definition, I present a Bayesian network based method to derive prior distributions for relatively complex models. The method borrows the Bayesian network model approach of using a directed acyclic graph to summarize our knowledge on the subject of interest and extends the Bayesian network to accommodate continuous variables. Continuous variables can enter the network through empirical models based on exploratory data analysis through existing models. The continuous variable Bayesian network modelling approach is illustrated using three examples – a model for evaluating the risk of Cryptosporidium contamination in US drinking water systems, model -based nutrient criteria for small rivers and streams in Ohio, and assessing water availabi lity to meet the use of both societal and ecological needs in the southeastern US.