Credibility Theory: Full Bayesian and Empirical Approaches

Full Bayesian Credibility

Full Bayesian credibility derives the posterior distribution of a risk parameter given observed data. For a Poisson-Gamma model (claim count Poisson with rate Lambda, where Lambda ~ Gamma(alpha, beta)), the posterior of Lambda given n observations is Gamma(alpha + sum(x_i), beta + n). The Bayesian premium is E[Lambda | data] = (alpha + sum(x_i))/(beta + n), a credibility-weighted average of the prior mean alpha/beta and the observed mean x-bar. The credibility weight Z = n/(n + beta) increases with data volume.

Empirical Bayes Approaches

Empirical Bayes methods estimate the prior distribution parameters from the data itself rather than specifying them subjectively. The nonparametric empirical Bayes approach (Buhlmann) uses within-group and between-group variation to estimate the credibility factor k = v/a without assuming a specific prior. The semiparametric approach assumes a parametric prior family but estimates its parameters from the collective experience. Both approaches yield credibility premiums of the form Z*x-bar + (1-Z)*mu, but differ in how Z and mu are estimated. Exam MAS-I tests both frameworks and their connections.