I am using a multilevel model constructed using stan to make predictions. I want to evaluate the model by making density predictions and evaluate those using the predictive likelihood for out-of-sample prediction y_{i,t+x} (also desirable for in-sample predictions) but I do not completely know how to compute these using stan or the stanfit output.

Currently, I construct point predictions by using the posterior parameter draws from the stanfit object and compute \hat{y}_{ij,t+x} = \frac{1}{M}\sum_{m=1}^M \mathbb{E}(y_{ij,t+x}|\boldsymbol{y}), with \boldsymbol{y} the values of y_{it} for t = 1,...,t. For example, when my dependent variable y_{it} is normally distributed with mean \alpha_{i} + \beta_{i}x_{it}, I can compute \hat{y}_{ij,t+x} = \frac{1}{M}\sum_{m=1}^M \alpha^{(m)}_{i} + \beta^{(m)}_{i}x_{i,t+x} as a point prediction. However, I do not know how to compute the posterior predictive likelihood, p(y_{i,t+x}|\boldsymbol{y}) for this observation. Does someone know how to do this? Or is there some other approach to compute this? For example, I do compute the replicated (in-sample) y_{it} values using this:

```
generated quantities{
real y_rep[Ni];
vector[Ni] log_lik;
for (i in 1:Ni) {
real eta_n = mu[i];
y_rep[i] = normal_rng(mu, sigma_e0);
log_lik[i] = normal_lpdf(y[i]| mu, sigma_e0);
}
}
```

Any help is really appreciated!