# Practical implementation of posterior predictive distribution

I'm trying to compute the posterior predictive distribution for a problem from samples of a Metropolis Hastings run. So as I understand the formula is \begin{equation} p(y^{new} | y) = \int_u p_{lik}(y^{new} | u) p_{post}(u | y) du, \end{equation} where $p_{post}$ is the posterior p.d.f and $p_{lik}$ the likelihood.

My first question is, do I really need the actual p.d.f of the posterior here? In my problem it is not analytically known, i.e. I only have the proportionality formula $p_{post}(u | y) \propto p_{lik}(y | u) p_{prior}(u)$. That is, I only have the unnormalized posterior. Can I just plug this into the above formula for the predictive?

Also, I'm thinking that, given the MCMC samples, lets say $(u_i), i=1, \ldots, n$, the integral should somehow be solved by Monte Carlo Integration, i.e. \begin{align} p(y^{new} | y) &= \int_u p_{lik}(y^{new} | u) p_{post}(u | y) du \\ &= \frac{1}{n} \sum_i^n p_{lik}(y^{new} | u_i) p_{post}(u_i | y) \end{align} Is that right? It seems somehow redundant that I already have samples form the posterior, $u_i$, and I evaluate the posterior p.d.f again...

And, finally, I'm wondering what exactly $y^{new}$ is. My understanding is that I chose a set of $y^{new}_i$ within some reasonable interval where I would expect new data points to lie.. then for each $i$ compute $p(y_i^{new} | y)$, which results in a number that tells how likely $y_i^{new}$ is. I'm not sure if I understood the concept of 'sampling from the predictive distribution' correctly, though...

Given that they are random draws from the posterior, you don't need $p_{post}(u_i|y)$ in your calculations; the probability of each of the $u_i$ you've drawn is now $1/n$, where $n$ is the number of M-H samples. $p(y^{new}|y)$ is approximated by $1/n \sum p(y^{new}|u_i)$. $p(u_i|y)$ was implicitly taken into account when drawing the M-H samples, so does not need to be included explicitly in the calculation of $p(y^{new}|y)$.
$y^{new}$ is just whatever value or values you want to calculate a posterior predictive probability for, nothing more than that.