Unexpected results from posterior predictive draws

I’m relatively new to stan and have a question regarding posterior predictive draws. I fitted the following hierarchical linear regression model and used the generated quantities block for posterior predictive draws. However, when I plot the densities of y and yrep, they differ a lot. Since I used N(0,1) as a prior for my parameters, I’m assuming this falls into the informative prior category and the model should be quite good at predicting outcomes.

data {
  int<lower=0> N;
  int<lower=0> K;
  int group[N];
  vector[N] y;
  vector[N] x;

parameters {
  vector[K] mu;
  vector[K] beta;
  real<lower=0> sigma;
  real nu_mu;
  real nu_beta;
  real<lower=0> tau_mu;
  real<lower=0> tau_beta;

model {
  nu_mu ~ normal(0,1);
  nu_b ~ normal(0,1);
  tau_mu ~ gamma(1,1);
  tau_b ~ gamma(1,1);
  sigma ~ gamma(1,1);
  mu ~ normal(nu_mu, tau_mu);
  beta ~ normal(nu_b, tau_b);
  for(n in 1:N)
    y[n] ~ normal(mu[group[n]] + beta[group[n]]*x[n], sigma);
//posterior predictive draws 
generated quantities {
   vector[K] beta_pred;
   vector[K] mu_pred;
   vector[N] y_rep;
   for (j in 1:Ngroup){
     mu_pred[j] = normal_rng(nu_mu, tau_mu);
   for (j in 1:Ngroup){
     beta_pred[j] = normal_rng(nu_b, tau_b);
   for (n in 1:N){
     y_rep[n] = normal_rng(mu_pred[group[n]] + beta_pred[group[n]]*x[n], sigma);
//posterior predictive density plots
y_pred1 <-as.matrix(fit.4, pars="y_rep")
ppc_dens_overlay(rep(y,20), y_pred1[1:100,])

Just to be clear, the dimension of y_pred1 is 4000, 2000. 2000 because K=20 and N=100

Screen Shot 2021-01-31 at 12.22.08 PM

Hi, the plot indicates to me that you have very poor fit, i.e., the model is not doing a good job of capturing the underlying data-generation model. There can be many reasons for this: use of wrong likelihood, wrong priors (too strong or too weak), combination of data+likelihood gives a strange posterior for HMC to explore, etc.

Can you provide data (or fake data) that shows this behavior, with some descriptions of the outcome and the predictors?