Sorry! Bad notation... What I meant was that we can’t compute the conditional posterior predictive density where $D = {(x_{1}, y_{1}), \dots, (x_{n}, y_{n})}$ . We can compute $p (~ y | ~ x, D, M)$ , where $M$ is some model, approximately using MCMC by drawing samples from the parameter space of $M$ , i.e. we can approximate the integral below using MCMC:

$p (~ y | ~ x, D, M) = \int_{θ \in Θ} p (~ y | ~ x, M, θ) p (θ | M, D) d θ$

where $Θ$ is the parameter space of $M$ . But the quantity that we are interested in is

$p (~ y | ~ x, D)$ not $p (~ y | ~ x, D, M)$ for a specific model i.e. we need to marginalise over the unknown model. How can we do this?