Define a random variable $X \in \mathbb{R}$. Suppose that the distribution of $X$ is dominated by the Lebesgue measure and hence admits a density (pdf) $f^*(x)$ at each $x \in \mathbb{R}$.
Is it possible to define $f^*$ variationally? For example, does there exist a function class $\mathcal{F}$ and a loss $L$ such that $f^* = \arg\min_{f \in \mathcal{F}} E[L\{X, f(X)\}]$? I'd prefer an (e.g.) $L$ to be as well behaved as possible.
I've searched through books and papers online and have not had any luck towards resolving my question.