  Accuracy and interpretability of a (non-life) insurance pricing model are
essential qualities to ensure fair and transparent premiums for policy-holders,
that reflect their risk. In recent years, the classification and regression
trees (CARTs) and their ensembles have gained popularity in the actuarial
literature, since they offer good prediction performance and are relatively
easily interpretable. In this paper, we introduce Bayesian CART models for
insurance pricing, with a particular focus on claims frequency modelling.
Additionally to the common Poisson and negative binomial (NB) distributions
used for claims frequency, we implement Bayesian CART for the zero-inflated
Poisson (ZIP) distribution to address the difficulty arising from the
imbalanced insurance claims data. To this end, we introduce a general MCMC
algorithm using data augmentation methods for posterior tree exploration. We
also introduce the deviance information criterion (DIC) for the tree model
selection. The proposed models are able to identify trees which can better
classify the policy-holders into risk groups. Some simulations and real
insurance data will be discussed to illustrate the applicability of these
models.
