This may be an elementary question. Say we have a univariate continuous random variable $X$ with unknown pdf $p(x)$, $ \forall x \in X$.

Presumably we can assign a second probability measure to the set of all possible values of the pdf $p$, define a Radon-Nikodym derivative etc.

Do these types of "hyper-distributions" have a general name? I do realize you can define distribution parameters probabilistically, I am looking for something more non-parametric.

P.S. Please let me know if my notation is incorrect!

This may be an elementary question. Say we have a univariate continuous random variable $X$ with unknown pdf $p(x)$, $ \forall x \in X$.

Presumably we can assign a second probability measure $Q(p)$ to the set of all possible realizations of the pdf $p$, define a Radon-Nikodym derivative etc.

Do these types of "hyper-distributions" have a general name? I do realize you can define distribution parameters probabilistically, I am looking for something more non-parametric.

P.S. Please let me know if my notation is incorrect!

Edit: There are examples of such distributions (see e.g. https://en.wikipedia.org/wiki/Dirichlet_process), but I am looking for a more general theory.

In case it helps, below is an example:

This is a simplified discrete case. Say we have two pdf's $p_1$ and $p_2$, where each has probability 0.5 under $Q$. While you can easily marginalize out the index (i.e. the $i \in \{1, 2\}$), I am interested in the distribution of possible pdf's.