Predictive measures of the conflict between frequentist and Bayesian estimators

• We propose a measure of discrepancy D between frequentist and Bayesian estimators.
• We find the predictive distribution of D (exponential families, conjugate priors).
• We show the effects of increasing sample size on predictive distribution of D.
• Explicit results/examples for Bernoulli and normal models are derived and discussed.
• We implement non-conjugate examples including application to the logistic models.

In the presence of prior information on an unknown parameter of a statistical model, Bayesian and frequentist estimates based on the same observed data do not coincide. However, in many standard parametric problems, this difference tends to decrease for growing sample size. In this paper we consider as a measure of discrepancy (Dn) the squared difference between Bayesian and frequentist point estimators of the parameter of a model. We derive the predictive distribution of Dn for finite sample sizes in the case of a one-dimensional exponential family and we study its behavior for increasing sample size. Numerical examples are illustrated for normal models.