data{
  int<lower=0> N;
  vector[N] x;
  vector<lower=0>[2] phi; //hyperparamter for mixing ratio
  real<lower=0> beta; //hyperparameter for center
}

parameters{
  simplex[2] ratio;
  real mu;
}

/**
calculate p(x_i,y_i=k|w) for each k
**/
transformed parameters{
  matrix[N,2] ln_p_xy;
  for(i in 1:N){
    ln_p_xy[i,1] <- log(ratio[1]) + normal_log(x[i],0,1);
    ln_p_xy[i,2] <- log(ratio[2]) + normal_log(x[i],mu,1);
  }
}

model{
  vector[2] sum_part;
  
  //priors
  ratio ~ dirichlet(phi);
  mu ~ normal(0,beta);
  
  //log likelihood
  for(i in 1:N){
    increment_log_prob(log_sum_exp(ln_p_xy[i]));
  }
}

/**
calculate p(y_i=k|x_i,w_j) k=1,2

we can obtain p(y_i=k|x_i, x^n) by sum_{j=1} p(y_i=k|x_i,w_j) / #(w_j)
on our interface for stan
**/
generated quantities{
  matrix[N,2] p_yx;
  for(i in 1:N){
    p_yx[i] <- softmax(ln_p_xy[i]')';
  }
}