I'm having some trouble following the logic of this passage from Chapter 14 in Bayesian Data Analysis, A. Gelman:
The numerical 'data' in a regression problem includes both $X$ and $y$. Thus, a full Bayesian model includes a distribution for $X$, $p(X|\psi$), indexed by a parameter vector $\psi$, and thus involves a joint likelihood $p(X,y|\psi,\theta)$, along with a prior distribution, $p(\psi,\theta)$. In the standard regression context, the distribution of $X$ is assumed to provide no information about the conditional distribution of $y$ given $X$; that is, we assume prior independence of parameters $\theta$ determining $p(y|X,\theta)$ and the parameters $\psi$ determining $p(X|\psi).$
Thus, from a Bayesian perspective, the defining characteristic of a 'regression model' is that it ignores the information supplied by $X$ about ($\psi$, $\theta$). How can this be justified? Suppose $\psi$ and $\theta$ are independent in their prior distribution; that is $p(\theta,\psi) = p(\theta)p(\psi)$. Then the posterior distribution factors,
$p(\psi,\theta|X,y) = p(\psi|X)p(\theta|X,y)$, [...]
When I work this out I can't obtain the last line. I can get
$p(\psi,\theta|X,y) = p(\psi|X,y,\theta)p(\theta|X,y)$.
Intuitively the statement makes sense, but I can't prove to myself that it is true.