Suppose I have a Bayesian hierarchical regression model where

Yi,j1,...,mindependentNormal(1,j+2,jxi,j+3,jxi,j2,2)

and

1,...,m,Normal3(,)

Let's say I have run my MCMC and I have posterior samples of beta.sim, theta.sim, sigma2.sim and Sigma.sim, where;

beta.sim is the 1,...,m samples

theta.sim is the samples

sigma2.sim is the 2 samples and

Sigma.sim is the samples.

How do I sample the posterior predictive of y~ given x~ (a vector of regressors). Do I use theta.sim and Sigma.sim to sample ~ first and then use ~ and sigma.sim to sample y~ ? OR do I directly sample y~ from theta.sim and sigma.sim

The question in this link below might be helpful

https://stats.stackexchange.com/questions/233343/generate-posterior-predictive-distribution-at-every-step-in-the-mcmc-chain-for-a