Quantifying the Information of the Prior and Likelihood in Parametric Bayesian Modeling

We suggest using a pair of metrics which quantify the extent to which the prior and likelihood functions influence inferences of parameters within a parametric Bayesian model, one of which is closely related to the reference prior of Berger and Bernardo. Our hope is that the utilization of these metrics will allow for the precise quantification of prior and likelihood information and mitigate the use of potentially nebulous terminology such as “informative”, “objectivity”, and “subjectivity”. We develop a Monte Carlo algorithm to estimate these metrics and demonstrate that they possess desirable properties via a combination of theoretical results, simulations, and applications on public medical data sets. While we do not suggest a default prior or likelihood, we suggest a way to quantify the information of the prior and likelihood functions utilized in a parametric Bayesian model; hence these metrics may be useful diagnostic tools when performing a Bayesian analysis.