  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.
