In section $4.4$ The Bayes Estimator of Alpaydin he discusses the use of the prior density of $p(\theta)$ to construct a posterior density for $\theta$. This is standard Bayesian estimation to get a parameter density.
He then goes on to say
For estimating the density at $\chi$, we have
$\begin{align} p(\chi | X) &= \int p(\chi, \theta | X) d\theta \\ &= \int p(\chi | \theta, X) p(\theta | X) d\theta \\ &= \int p(\chi | \theta) p(\theta | X) d\theta \end{align}$
At this point is he still talking about the posterior density of the parameter evaluated at a given $\chi$ or is he talking about the actual density of our data $p(\chi)$?