fluctuation theory is to use confidence intervals to deter-
mine thresholds above which a data set is “fully credible.”
Hans Bühlmann pointed out the dubious logic of assigning
full credibility to a data set based on a confidence interval
that, by definition, covers the true value with a probability of
less than one. Outside the United States, the global actuarial
community refers to limited fluctuation credibility as “Ameri-
can credibility theory.” This perhaps reflects a widely shared
point of view that credibility is best interpreted as a funda-
mentally Bayesian, not frequentist, statistical theory.
Emphasizing proper foundations can yield significant prac-
tical benefits. Before the introduction of generalized linear
models (GLM), actuaries used purely algebraic Bailey-Simon
methods for multivariate insurance rating problems. Once it
was realized that the Bailey-Simon methods were in effect
special cases of GLM, the actuarial community vigorously
embraced the use of GLM. This not only has made ratemak-
ing more statistically rigorous; it also has broadened the range
of actuarial practice by providing actuaries with a powerful
mainstream statistical tool for attacking a variety of problems.
Given the recent proliferation of MCMC simulation-based
methodologies—and the software to implement them—for
solving Bayesian problems, we foresee a similar evolution from
ad hoc credibility formulas to a more widespread use of Bayes-
ian hierarchical models in coming decades.
■ a Unifying framework—This discussion can be taken a
step further: The “empirical Bayes” theory of multilevel/
hierarchical models in fact unifies the standard GLM-based
ratemaking framework and the Bühlmann-Straub credibility
theory. Hierarchical models enable one to include highly cat-
egorical dimensions (such as vehicle make/model, territory,
or business classification) in GLM models in a way that avoids
estimating too many parameters using too little data. This is
achieved through the same “pooling” and “shrinking” process
embodied in Bühlmann credibility. (See Andrew Gelman and
Jennifer Hill’s textbook, Data Analysis Using Regression and
Multilevel/Hierarchical Models, and James Guszcza’s article
in the Fall 2008 issue of the Casualty Actuarial Society Forum,
“Hierarchical Growth Curve Models for Loss Reserving,” for
an elaboration on these comments.) Today Bayesian credibil-
ity theory and GLM are viewed as separate cornerstones of
actuarial practice. But they need not be viewed as distinct
subjects: Both are special cases of multilevel models.
