Sample Questions
Question 1
Easy
Which of the following best characterizes the relationship between Buhlmann credibility and Bayesian credibility?
Solution
B is correct. Buhlmann credibility is the best linear unbiased estimator of the Bayesian posterior mean in the mean-squared-error sense. It restricts the estimator to be an affine function of the observed sample mean and the prior mean and chooses the weights to minimize expected squared error. In general the true Bayesian posterior mean is nonlinear in the data, and Buhlmann's linear form is only an approximation. The two coincide exactly when the posterior mean happens to be linear in the data, as in conjugate exponential-family pairs. So the correct characterization is that Buhlmann is the linear least-squares approximation to the Bayesian posterior mean.
Question 2
Medium
Compared with a fully Bayesian credibility estimate, the Buhlmann credibility estimate is most accurately characterized as which of the following?
Solution
A is correct. Buhlmann credibility is derived as the linear function of the observations (and the manual rate) that minimizes mean squared error relative to the Bayesian posterior mean. It is therefore the best linear unbiased predictor of the underlying hypothetical mean and equals the Bayesian estimate exactly when the Bayesian estimate is itself linear in the data (for example, Poisson-Gamma, Normal-Normal, and Bernoulli-Beta conjugate pairs). Otherwise it is a linear approximation rather than a generally smaller, larger, or transformed quantity.
Question 3
Hard
Buhlmann credibility and Bayesian credibility coincide exactly under certain model specifications. Which of the following pairings of likelihood and prior produces this exact equivalence?
Solution
E is correct. Bayesian and Buhlmann credibility coincide exactly when the Bayesian posterior mean is itself a linear function of the observed mean. This linearity arises in the standard conjugate exponential-family models, of which the Poisson likelihood with a Gamma prior is the canonical example. Other conjugate combinations with the same exact-linearity property include Normal/Normal, Bernoulli/Beta, and Exponential/Gamma. In each, the posterior mean can be written as a credibility-weighted average of the prior mean and the observed mean with weights matching the Buhlmann formula. The non-conjugate pairings listed elsewhere lack this closed-form linear posterior, so Buhlmann is only an approximation rather than an exact match.
