5.1 Posterior predictive checks

We have previously introduced and discussed posterior predictive checks as a way to assess how well a model explains the data used to fit a model. The purpose of this type of testing is not to determine whether a model is incorrect; we already know this! The goal of the exercise is to understand how well we are capturing the data. By performing posterior predictive checks, we aim to better understand the limitations of a model. Once we understand the limitations, we can simply acknowledge them or try to remove them by improving the model. It is expected that a model will not be able to reproduce all aspects of a problem and this is usually not a problem as models are built with a purpose in mind. As different models often capture different aspects of data, we can compare models using posterior predictive checks.

Let’s look at a simple example. We have a dataset with two variables, x and y. We are going to fit these data with a linear model:

We...