How to find the marginal prior distribution?

Question

Suppose that $\beta$ has the following prior

$$ \beta|\zeta \sim f(\beta,\zeta) $$

Then I know that the marginal prior distribution of $\beta$ is given by

$$ \int f(\beta,\zeta) d\zeta $$

However, suppose that we have the following hierarchical prior model

$$ \beta|\zeta \sim f(\beta,\zeta) $$ $$ \zeta|\eta \sim g(\zeta,\eta) $$ $$ \eta \sim h(\eta) $$

Then what is the marginal prior distribution of $\beta$?

Answer 1

If we further assume that $\beta \ \bot \ \eta | \zeta$ you have the conditional density $p(\beta|\zeta,\eta) = p(\beta|\zeta) = f(\beta,\zeta)$, which then lets you expand the joint density as:

$$\begin{align} p(\beta,\zeta,\eta) &= p(\beta|\zeta,\eta) \ p(\zeta|\eta) \ p(\eta) \\[6pt] &= f(\beta,\zeta) \ g(\zeta,\eta) \ h(\eta). \\[6pt] \end{align}$$

You can then write the marginal density of $\beta$ as:

$$p(\beta) = \int \int f(\beta,\zeta) \ g(\zeta,\eta) \ h(\eta) \ d\zeta \ d\eta.$$

Note that when practitioners write out hierarchical models like this, the conditional independence conditions for this kind of expansion are usually taken to be implicit, so as to give full specification of a joint density. So here we would take the condition $\beta \ \bot \ \eta | \zeta$ to be implicit in the specification of the model, unless there were some compelling reason to think it is not intended (in which case the joint density is not fully specified).