Part 6: The bridge to Bayesian — what Bühlmann-Straub assumes
Part 6: The bridge to Bayesian — what Bühlmann-Straub assumes¶
B-S is empirical Bayes¶
Bühlmann-Straub is an empirical Bayes method. This is rarely stated clearly in actuarial textbooks, but it motivates everything in the second half of this module.
The credibility premium P_i is exactly the posterior mean of this Bayesian model:
X_{ij} | theta_i ~ Normal(theta_i, v / w_{ij}) [observation model]
theta_i ~ Normal(mu, a) [prior on group mean]
The posterior mean of theta_i is:
where Z_i = w_i / (w_i + K), K = v/a. That is exactly the Bühlmann-Straub formula.
B-S plugs in point estimates of v and a (the structural parameters we estimated above). Full Bayesian treats v and a as uncertain — it places priors on them and integrates over that uncertainty. Three things follow from this:
When B-S is sufficient: - Many groups (20+) so that v and a are estimated reliably from data - One grouping variable - Results needed in seconds, not minutes - Regulatory documentation needs to be simple and traceable
When full Bayesian is better: - Few groups (fewer than 10 affinity schemes) — with few groups, the estimate of the between-group variance is unreliable, and that uncertainty propagates into Z. Full Bayesian propagates it correctly; B-S ignores it. - Multiple crossed grouping variables simultaneously (area AND vehicle group AND NCD band) - Proper Poisson or Gamma likelihood — B-S assumes Normal errors in its derivation - Credible intervals on individual segment rates are required for regulatory evidence or pricing decisions on thin segments