data {
  int N;
  vector[N] y;
  vector[N] X;
}
parameters {
  real a;
  real b;
  real<lower=0> sigma;
}
model {
  target+= normal_lpdf(y|a+X*b, sigma);     // data model
  target+=(-log(sigma));      // log prior for p(sigma) propto 1/sigma
}
generated quantities {
  vector[N] log_lik;          // pointwise log-likelihood
    for (n in 1:N){
        log_lik[n] = normal_lpdf(y[n]| a+X[n]*b, sigma);
  }
}