Bayesian Credibility for GLMs

TL;DR

This paper introduces a Bayesian approach to deriving credibility premiums for generalized linear models (GLMs), allowing flexible prior choices and non-linear premiums, demonstrated through real data and simulations.

Contribution

It extends classical credibility theory by incorporating modern Bayesian methods with flexible priors and non-linear premiums, using relative entropy as a loss function.

Findings

Feasibility demonstrated on real insurance data

Flexible prior distributions can be used without conjugacy restrictions

Non-linear credibility premiums can be derived using the proposed approach

Abstract

We revisit the classical credibility results of Jewell and B\"uhlmann to obtain credibility premiums for a GLM using a modern Bayesian approach. Here the prior distributions can be chosen without restrictions to be conjugate to the response distribution. It can even come from out-of-sample information if the actuary prefers. Then we use the relative entropy between the "true" and the estimated models as a loss function, without restricting credibility premiums to be linear. A numerical illustration on real data shows the feasibility of the approach, now that computing power is cheap, and simulations software readily available.