Logit simplex

[spinkney]

vector log_logistic_simplex_lp(vector stick_slices) {
     int K = num_elements(stick_slices) + 1;
     vector[K] log_pi;
    
     real log_stick = 0;
     for (k in 1:K - 1) {
       real log_inv_logit_stick = log_inv_logit(stick_slices[k]);
       log_pi[k] = log_inv_logit_stick  + log_stick;
       log_stick = log_diff_exp(log_stick, log_pi[k]);
       // the jacobian for inv_logit is
       // target += log_inv_logit(y) + log1m_inv_logit(y);
       // because this is log_inv_logit(y)
       // we can use the chain rule 
       // jacobian for f'(y) = d log_inv_logit(y) / dy
       // = d log(inv_logit(y))/d log(inv_logit(y)) + d inv_logit(y) / dy 
       // = -log_inv_logit(y) + log_inv_logit(y) + log1m_inv_logit(y) 
       // = log1m_inv_logit(y) 
       target += log1m_inv_logit(stick_slices[k]);
       target += log_stick + log1m_exp(log_stick) + log1m_exp(log_pi[k]);
     }
     
     log_pi[K] = log_stick;
     
     return log_pi;
   }