# How does the marginal distribution become the prior distribution?

Bayes' Theorem for densities/pmf's states that, given, say, two univariate random variables $x,z$ we have

$$p(z\mid x) = \frac {p(x\mid z)\cdot p(z)}{p(x)}$$

This is part of the core of the probability theory most widely used nowadays.

But in order to use this theoretical result for Bayesian statistical inference, two basic mappings are apparently made:

• First, $p(x\mid z)$, goes from being a conditional density to "the likelihood of the sample", a step also used in the maximum likelihood method.

• Second, $p(z)$, goes from being the marginal density to the "prior" density of $z$.

I am interested in this second mapping: can somebody lay out or suggest some literature where this step of going from a marginal density to a "prior" density is discussed in either technical or philosophical detail?

I’m guessing any examples from the literature would be rather "old,” because after decades of Bayesian statistics, calling $p(z)$ the "prior density" has become self-evident.

But to write "$p(z)$ is the prior density that incorporates all the prior beliefs/knowledge about $z$ except the sample at hand" raises the question: "where is the marginal density of a random variable defined in such a way? And if it is not defined in such a way (and it is not), then, what are the arguments that assert that the marginal density is indeed the proper object to represent what we want the "prior density" to represent?"

• When you say "cannot be directly observed" how do you deal with different people "observing" different things? Also, the posterior $p(x|z)$ (ie infer $x$ given observed $z$) is just as valid mathematically as $p(z|x)$ is it not? – probabilityislogic Jun 27 '20 at 4:55