  The claim experience of the past is a very important information to calculate
the fair price of an insurance contract. In a lot of European countries for
instance the prices for motor car insurance depend on the number of claims the
driver has reported to the insurance company during the last years. Classically
these prices are calculated on the basis of a mixed Poisson model with a gamma
mixing distribution. The mixing distribution models the car drivers' qualities
across the insured portfolio. This is just one example for experience rating.
In the classical context the price is equal to the expectation of the Bayesian
posterior distribution.
  In some lines of business (especially third party liability and lines with
exposure to extreme weather events) we that the real world data cannot be
described well enough by the classical Poisson - gamma model. Therefore we
investigate the influence of the mixing distribution on the posterior
distribution conditional on the experienced number of claims. This enables the
application of other - more risk adequate premium principles than the
expectation principle. We introduce the inverse - gamma distribution as a new
mixing distribution to model claim numbers and compare it to the classical
gamma distribution. In both cases a closed analytic representation of the mixed
distribution can be found: In the classic case the well known negative binomial
distribution, in our new one a representation using the Bessel functions.
Additionally we present numerical results about the tail behaviour of the mixed
Poisson - inverse - gamma distribution. Finally we introduce the concept of
resolution. It enables us to decide if the classification of risk groups via
the number of experienced claims is a risk adequate procedure.
