A normative definition of a Bayesian prior

Applications of the Bayesian statistics require specifying a prior distribution for each unknown parameter to be estimated. The commonly used definition of a Bayesian prior distribution, information about an uncertain parameter, does not provide guidance on how to derive and formulate a prior distribution. In practice, we often use "non-informative" priors or priors based on mathematical convenience. I present a normative definition of the prior based on the shared features of the James-Stein estimator, the empirical Bayes method, and the Bayesian hierarchical model. I use the word "normative" to mean "prescriptive". It also reflects the meaning that the definition can be inconsistent with one another insofar as different types of parameters. I present two case studies where this definition guided me to formulate the modeling processes: one on modeling and predicting cyanobacterial toxin concentration in Lake Erie using chlorophyll-a concentrations (Lake Erie example) and the other on improving the accuracy of calibration-curve-based chemical measurement method (calibration-curve example). The Lake Erie example illustrates temporal exchangeable units, while the calibration-curve example showcases the ubiquity of such exchangeable units.