  This article, in a first step, considers two Bayes estimators for the
relativity premium of a given Bonus--Malus system. It then develops a linear
relativity premium that closes, in the sense of weighted mean square error
loss, to such Bayes estimators. In a second step, it supposes that the claim
size distribution for a given Bonus--Malus system can be formulated as a finite
mixture distribution. It then evaluates the base premium under a Bayesian
framework for such a finite mixture distribution. The Loimaranta efficiency of
such a linear relativity premium, for several Bonus--Malus systems, has been
compared with two Bayes and ordinary linear relativity premiums.
