There are two ways you might think of ABC. One interpretation is that it is, as the name suggests, an approximation that is faster to compute than the exact value.
But remember that Bayesian analysis is always based on modeling decisions, which implies that there is no “exact” solution. For any interesting physical system there are many possible models, and each model yields different results. To interpret the results, we have to evaluate the models.
So another interpretation of ABC is that it represents an alternative model of the likelihood. When we compute , we are asking “What is the likelihood of the data under a given hypothesis?”
For large datasets, the likelihood of the data is very small, which is a hint that we might not be asking the right question. What we really want to know is the likelihood of any outcome like the data, where the definition of “like” is yet another modeling decision.
The underlying idea of ABC is that two datasets are alike if they yield the same summary statistics. But in some cases, like the example in this chapter, it is not obvious which summary statistics to choose.