As far as I know, random variables are functions form a probability space $(\Omega,\mathcal{A},\mu)$ to real numbers $\mathbb{R}$, i.e. $X:\Omega\to\mathbb{R}$. Let a probability density function exist $f_{X}$ and let it depend on an unknown parameter $c\in\mathbf{c}$ where $\mathbf{c}$ is set of all possible values of the parameter.
Bayesian inference, as far as I know works with some prior probability density function for $c$, i.e. $f_C$ and generates using the Bayes rule some posterior $f_{C|X}$ with respect to realization of $X$ that is denoted as $x$. My questions are:
- What is the parameter $C$? Is it considered to be a random variable if it has probability density function $f_{C}$?
- If so, what is its relationship to $(\Omega,\mathcal{A},\mu)$? Does it mean that $C:\Omega\to\mathbb{R}$?
The answer might be just yes/no, possibly with some appropriate reference.
Yes, $C$ is a random variable with PDF $f_C$ (when $C$ has a density).
Yes, $C$ is defined on $\Omega$, if only to be able to consider objects such as the conditional distribution of $X$ conditionally on $C$, and no, $C$ is not always real valued.