data { 
  int<lower=1> N; // number of data points
  vector[N] z;    // observations

} 
parameters {
  real<lower=0> mu;
  simplex[2] theta;
}
model {
  vector[2] log_theta = log(theta);
  
  mu ~ normal(0,5);
  

  for(n in 1:N){
    vector[2] lps = log_theta;
    lps[1] += normal_lpdf(z[n] |  0, 1) - normal_lccdf(1.96 |  0, 1);
    lps[2] += normal_lpdf(z[n] | mu, 1) - normal_lccdf(1.96 | mu, 1);
    
    target += log_sum_exp(lps);
  }

  
}