From a tutorial a came across, the author has two likelihood statements in the model block; I’m curious if this is (1) incorrect, (2) works but not the cleanest/best way to specify a model or (3) the right way to do things in Stan.

The Stan code:

```
data {
int n; // number of observations
int n_county; // number of counties
vector[n] log_radon;
vector[n] vfloor;
int<lower = 0, upper = n_county> county[n];
}
parameters {
vector[n_county] alpha; // vector of county intercepts
real beta; // slope parameter
real<lower = 0> sigma_a; // variance of counties
real<lower = 0> sigma_y; // model residual variance
real mu_a; // mean of counties
}
model {
// conditional mean
vector[n] mu;
// linear combination
mu = alpha[county] + beta * vfloor;
// priors
beta ~ normal(0, 1);
// hyper-priors
mu_a ~ normal(0, 1);
sigma_a ~ cauchy(0, 2.5);
sigma_y ~ cauchy(0, 2.5);
// level-2 likelihood
alpha ~ normal(mu_a, sigma_a);
// level-1 likelihood
log_radon ~ normal(mu, sigma_y);
}
generated quantities {
vector[n] log_lik; // calculate log-likelihood
vector[n] y_rep; // replications from posterior predictive distribution
for (i in 1:n) {
// generate mpg predicted value
real log_radon_hat = alpha[county[i]] + beta * vfloor[i];
// calculate log-likelihood
log_lik[i] = normal_lpdf(log_radon[i] | log_radon_hat, sigma_y);
// normal_lpdf is the log of the normal probability density function
// generate replication values
y_rep[i] = normal_rng(log_radon_hat, sigma_y);
// normal_rng generates random numbers from a normal distribution
}
}
```

I’m also unclear on why the author is defining log likelihood statements in the generating quantities portion. I had believed this section is primarily deterministic operations on the inferred variables and isn’t, itself, “part of the model”. Is this correct?