Bayesian credibility

There is a deeper way to see the whole credibility story. Before you observe a risk, you have a belief about its true mean — a prior, formed from the population of risks. Then the risk's actual experience arrives as evidence. Bayes' theorem tells you exactly how to update your belief into a posterior, and your new best estimate of the risk's true mean is the mean of that posterior. Bayesian credibility is simply credibility done this way: the estimate is the Bayesian posterior mean.

Remarkably, in many natural models the posterior mean comes out as the same Z-blend we have seen all along: Z times your own data plus (1 minus Z) times the prior mean, with Z rising as you gather more data. When the posterior mean is exactly a linear blend like this, the situation is called 'exact credibility', and Buhlmann's linear estimate coincides with the full Bayesian answer — no approximation lost. This happens for tidy conjugate pairs, such as a Poisson claim count with a gamma prior on its rate, or a normal mean with a normal prior. In those cases the Bayesian view and the Buhlmann view literally agree.

The Bayesian view is the conceptual home of credibility: it explains why blending toward a prior is not a fudge but the rigorously correct way to combine new evidence with prior knowledge. The honest caveat is twofold. First, you must commit to a prior, and a poorly chosen prior biases the answer (empirical Bayes eases this by estimating the prior from data). Second, outside the special conjugate cases the posterior mean is generally not a simple linear blend, so Buhlmann's tidy formula is then only an approximation to the true Bayesian estimate.