Empirical Bayes
Empirical Bayes estimates the prior distribution from the portfolio, then uses that fitted prior to shrink individual risks toward the collective pattern.
Quick Answer
Empirical Bayes sits between a fully specified Bayesian model and a purely frequentist fit. You use portfolio data to estimate the prior or credibility parameters, then use that estimated prior to calculate risk-level predictions.
The ASTAM syllabus explicitly names empirical Bayesian estimation in the nonparametric and semiparametric cases, and it also asks candidates to understand how Buhlmann models relate to Bayesian models.
Gamma-Poisson Example
Suppose a claim count N for one risk is Poisson with mean e theta, where e is exposure. Suppose theta has a Gamma(alpha, beta) prior. After observing y claims over exposure e, the posterior mean is a weighted blend of the observed rate y/e and the prior mean alpha/beta.
In empirical Bayes, alpha and beta are estimated from the portfolio instead of being handed to you. That is the empirical step: the portfolio supplies the prior shape.
Connection To Credibility
Buhlmann credibility can be read as an empirical Bayes approximation. It does not require the full prior distribution, but it still uses portfolio-level variance components to decide how much to trust each risk's own experience.
That is why credibility estimates often look like Bayesian posterior means even when the formulas are written in EPV and VHM language.
Worked Example
A portfolio fit gives alpha = 3 and beta = 2 for annual claim frequency per exposure unit. A risk with e = 4 exposure units has y = 10 observed claims. The posterior mean frequency is (3 + 10) / (2 + 4) = 13/6 = 2.167 claims per exposure unit.
The observed rate is 10/4 = 2.5. The prior mean is 3/2 = 1.5. The posterior mean lands between them because the individual data is useful but not final.
Exam Answer Shape
A strong written answer states the fitted prior assumption, shows the posterior or credibility calculation, and then gives a business interpretation. For counts, that usually means explaining how the expected claim frequency changes after seeing the risk's own experience.
A weak answer computes a number and stops. The actuarial point is the shrinkage decision: which signal moved the estimate, how much it moved, and what uncertainty remains.