Translating Predictive Distributions into Informative Priors

When complex Bayesian models exhibit implausible behavior, one solution is to assemble available information into an informative prior. Challenges arise as prior information is often only available for the observable quantity, or some model-derived marginal quantity, rather than directly pertaining to the (usually latent) parameters in our model. We propose a method for translating available prior information, in the form of an elicited distribution for the observable or model-derived marginal quantity, into an informative joint prior. Our approach proceeds given a parametric class of prior distributions with as yet undetermined hyperparameters, and minimizes the difference between the supplied elicited distribution and corresponding prior predictive distribution. We employ a global, multi-stage Bayesian optimization procedure to locate optimal values for the hyperparameters. Three examples illustrate our approach: a cure-fraction survival model, where censoring implies that the observable quantity is a priori a mixed discrete/continuous quantity; a setting in which prior information pertains to R2—a model-derived quantity; and a nonlinear regression model. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.