It is obvious that probability density does not necessarily exist for a given probability distribution. However, many statistical applications assume the existence of density and even use closed form density functions. For example, when calculating posterior distributions we usually use density functions of data and prior. In Neyman-Pearson school of hypothesis testing, likelihood ratio is used. There are many more examples. So I am wondering if the existence of probability densities is essential to statistics, in either theory or applications?
My background is mostly in probability and I have less experience with statistics. Maybe I miss a lot of commonly used statistical models, but I would really appreciate if anyone points that out.