density estimation

Given samples $X_1,\dots ,X_n\sim P$ we want to estimate the density of $P$. We can consider both parametric density estimation and nonparametric density estimation, the latter of course being easier but sensitive to model misspecification.

Density estimation is obviously a very common task. Once you have an estimate of the distribution, you can calculate regression functions, perform anomaly and outlier detection (just look at whether a new observation has a small probability) and hypothesis testing (in particular two-sample testing, in which we can estimate the densities of the two samples compute the divergence between them).