Measuring Bayesian sensitivity in the compound Poisson process

Abstract

Bayesian methods are widely used to determine insurance premiums, though they are sometimes criticized for the arbitrariness in selecting prior distributions. To mitigate this issue, classes of priors incorporating expert knowledge have been proposed, allowing for the analysis of uncertainty through upper and lower bounds on Bayesian premiums. In this paper, we employ a recently introduced class of priors based on stochastic orders, where the induced order on prior distributions is preserved in the corresponding posterior distributions. Uncertainty around a prior is captured through weighted functions, and the extremal elements of the class define premium bounds. We also show how dependence among parameters can be integrated using suitable weight functions. Our approach is developed within the framework of the compound Poisson process, a fundamental model for claim frequency and severity in car insurance. Additionally, we present a sensitivity analysis method for a bonus–malus system (BMS).