Age-Period-Cohort (APC) models are well used in the context of modelling health and demographic data to produce smooth estimates of each time trend. When smoothing in the context of APC models, there are two main schools, frequentist using penalised smoothing splines, and Bayesian using random processes with little crossover between them. In this article, we clearly lay out the theoretical link between the two schools, provide examples using simulated and real data to highlight similarities and difference, and help a general APC user understand potentially inaccessible theory from functional analysis. As intuition suggests, both approaches lead to comparable and almost identical in-sample predictions, but random processes within a Bayesian approach might be beneficial for out-of-sample prediction as the sources of uncertainty are captured in a more complete way.

翻译：年龄-时期-队列模型广泛应用于健康和人口数据建模中，用以生成各时间趋势的平滑估计值。在APC模型的平滑处理中，主要存在两大流派：频率学派采用惩罚平滑样条方法，而贝叶斯学派则运用随机过程方法，两者之间鲜有交叉。本文清晰阐述了两大流派之间的理论联系，通过模拟数据和真实数据实例展示其异同，并帮助APC模型的一般使用者理解原本可能难以触及的泛函分析理论。正如直觉所暗示，两种方法能得到高度相似、几乎完全相同的样本内预测结果；然而，由于贝叶斯框架下的随机过程能以更完整的方式捕捉不确定性来源，其在样本外预测中可能更具优势。