Thanks. This helps. Can you give a specific example of when someone would use this calculation, and where it would go wrong? Or what kind of problem requires the draws to be a function of all the data?
This is the core calculation to all of Bayesian statistics, so it's used pretty much everywhere. Almost any time you try to compute p(unknown|data) you run into computational issues. It's a high dimensional integration problem that has no closed form solution. There are models that can be simplified, but they are the minority.