Credibility Theory

A pricing problem requires a credibility weight Z based on volume and a desired precision level, and you must determine partial or full credibility.

For full credibility in pure premium method, n_F = (z_{α/2}/k)^2 (1 + CV^2_X) (or simpler forms for frequency-only or severity-only). For partial credibility, Z = √(n/n_F). The credibility-weighted estimate is Z × experience + (1 - Z) × complement of credibility.

n_F depends on (precision k, confidence α, process CV); Z = min(√(n/n_F), 1).

A portfolio is described by between-class and within-class variability and you must compute the Buhlmann credibility factor and the credibility-weighted premium.

Compute the expected process variance v = E[Var(X|θ)] and the variance of the hypothetical means a = Var(E[X|θ]). Set K = v/a. Then Z = n/(n + K) where n is the number of observations. Buhlmann-Straub adjusts Z when exposures vary across observations: Z = total exposure / (total exposure + K).

K = E[Var(X|θ)] / Var(E[X|θ]); Z = n / (n + K).

A risk parameter has a prior distribution (e.g., gamma for Poisson rate) and you must compute the posterior predictive premium.

Choose the conjugate pair: gamma-Poisson, beta-Bernoulli, normal-normal, gamma-exponential. The posterior parameters update simply from the prior and the sample. The posterior predictive mean is the credibility-weighted estimate, and it coincides with Buhlmann credibility when the linearity assumption holds.

Conjugate prior gives posterior in same family with updated hyperparameters; Bayes mean coincides with Buhlmann in exact-credibility families.