# Is it true that the word prior should be used only with latent random variables?

I use to think that any marginal distribution on a random variable can be called as prior probability distribution.

Is it true?

Or is it false due to the reason that the random variable under consideration may not be a latent random variable and the term prior should be used strictly with the marginal distributions over a latent random variable only?

In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether they are observed or hidden. Say for example if you believe $$X$$ follows a distribution defined by $$\theta$$, denote $$X \sim P(X|\theta)$$ Where $$\theta$$ is the parameter of the distribution, from Bayesian perspective it's also a random variable. Usually in this case random variable $$X$$ is observed and $$\theta$$ is not, and you want to infer/learn/esitmate $$\theta$$ based on your observations. In such situations there's no matter of "prior", "marginal" or "posterior"
The term "prior", "marginal" or "posterior" matters when you believe $$\theta$$ follows some other distribution $$\theta \sim P(\theta|\gamma)$$ Then we call this "other distribution" the prior, more specifically it's the piror distribution for $$\theta$$. Among all three random variables $$X$$, $$\theta$$ and $$\gamma$$, usually $$X$$ and $$\gamma$$ are observed, $$\theta$$ is not, and you want to estimate $$\theta$$ based on the observed $$X$$ and $$\gamma$$. So yes the term "prior" is usually on hidden random variables, of course you can believe there's a prior distribution for $$\theta$$ even when it is observed, but usually nobody do so(why would anyone esitimate something that is already observed?). And, if you can't observe $$\gamma$$, you can even assume $$\gamma$$ follows a distribution defined by another random variable $$\eta$$, then $$P(\gamma | \eta)$$ will be the prior for $$\gamma$$. Hope this answers your question regarding to "prior".
Now let's talk about "marginal". In previous example people usually interested in the distribution of $$X$$ (while $$\theta$$ is hidden), given $$\gamma$$, the distribution $$X \sim P(X|\gamma)$$ is called the "marginal distribution". The term "marginal" came from the fact that $$P(X|\gamma)$$ is acquired by marginalizing out $$\theta$$ from the joint distribution: $$p(X|\gamma) = \int_\theta p(X|\theta)p(\theta|\gamma)$$