Bayesian Fair Experience Premium Under Linear Exponential Likelihoods and Conjugate Priors

INTRODUCTION: In this chapter we will prove the point wise convergence of the Bayesian point estimates under conjugate priors belonging to the 'linear exponential families'. The result below demonstrates that if E[X q] =q , then the Bayesian Point estimates
(i) Have a generic formula that also provides the empirical Bayes point estimate of q .
(ii) Converge point-wise to the sample mean under increasing sample sizes, as do the Maximum Likelihood estimates.