Maximum Likelihood Densities

Why

We want to summarize a dataset in $\R $ with a density.

Definition

The likelihood (or density likelihood) of a density $f: \R \to \R $ on a datset $x^1, \dots , x^n \in \R $ is $\prod_{k = 1}^{n} f(x^k)$. A maximum likelihood density is a density which maximizes the likelihood among all densities.

As with probability distributions, we say that we are selecting the distribution according to the maximum likelihood principle. In general, we call any function from datasets to densities a density selector.