Buhlmann (greatest-accuracy) credibility

Classical credibility only asks 'is my data noisy?'. Hans Buhlmann's 1967 idea asks the sharper question: 'how much do risks genuinely differ, and how noisy is each one's data?' — and then chooses the weight Z that makes the estimate as accurate as possible. Instead of aiming merely to limit fluctuation, it directly minimizes the expected squared error of the prediction. That is why it is called greatest-accuracy credibility.

The model imagines that each risk has its own hidden true mean (its 'hypothetical mean'), drawn from a population of risks, and that its observed claims scatter around that hidden mean. Two quantities then matter: how much the hidden means differ between risks (the variance of the hypothetical means, VHM), and how much each risk's data bounce around its own mean (the expected process variance, EPV). Buhlmann shows the best linear estimate is the familiar blend with Z = n/(n + k), where k = EPV/VHM and n is the number of observations. Risks that differ a lot (big VHM, small k) earn high Z; when data are very noisy relative to real differences (big EPV, big k), Z stays low.

Buhlmann credibility is the theoretical backbone of modern experience rating. Its great virtue over classical credibility is that it answers 'why trust your data' with two estimable variances rather than an arbitrary fluctuation target, and it always blends toward the overall mean of the whole collective. Its honest catch: you must estimate EPV and VHM from data (the empirical-Bayes step), and the result is the best linear predictor, not necessarily the best of all possible predictors — for that you would need the full Bayesian posterior, which Buhlmann approximates.