A posteriori ratemaking in insurance uses a Bayesian credibility model to upgrade the current premiums of a contract by taking into account policyholders' attributes and their claim history. Most data-driven models used for this task are mathematically intractable, and premiums must be then obtained through numerical methods such as simulation such MCMC. However, these methods can be computationally expensive and prohibitive for large portfolios when applied at the policyholder level. Additionally, these computations become ``black-box" procedures as there is no expression showing how the claim history of policyholders is used to upgrade their premiums. To address these challenges, this paper proposes the use of a surrogate modeling approach to inexpensively derive a closed-form expression for computing the Bayesian credibility premiums for any given model. As a part of the methodology, the paper introduces the ``credibility index", which is a summary statistic of a policyholder's claim history that serves as the main input of the surrogate model and that is sufficient for several distribution families, including the exponential dispersion family. As a result, the computational burden of a posteriori ratemaking for large portfolios is therefore reduced through the direct evaluation of the closed-form expression, which additionally can provide a transparent and interpretable way of computing Bayesian premiums.

翻译：保险后验费率厘定使用贝叶斯可信度模型，通过考虑保单持有人的属性及其索赔历史来更新当前合同保费。大多数用于此任务的数据驱动模型在数学上难以处理，必须通过数值方法（如MCMC模拟）获取保费。然而，当应用于单个保单持有人层面时，这些方法对于大规模组合可能产生高昂的计算成本，甚至难以实施。此外，由于缺乏显示如何利用保单持有人索赔历史更新保费的表达式，这些计算过程成为"黑箱"操作。为解决这些挑战，本文提出采用代理建模方法，以低成本推导任意模型的贝叶斯可信度保费闭合表达式。作为方法论的一部分，本文引入"可信度指数"——该指标是保单持有人索赔历史的汇总统计量，作为代理模型的主要输入，且对包括指数分散族在内的多个分布族具有充分性。通过直接评估闭合表达式，大规模组合的后验费率厘定计算负担得以降低，同时该方法还能提供透明可解释的贝叶斯保费计算方式。