/**********************************************************************
 * heart transplant mortality rate 
 **********************************************************************/

data {
    int<lower=0> n;               // sample size 
    int<lower=0> death[n];        // number of death  
    real exposure[n];             // number of exposure 
    real<lower=0> z0;             // parameter in prior 
}
parameters {
    real<lower=0> mu;             // hyper-parameter 
    real<lower=0> alpha;          // hyper-parameter 
    real<lower=0> rate[n];        // mortality rate 
}
model {
    for(i in 1:n) death[i] ~ poisson(exposure[i] * rate[i]); 

    rate ~ gamma(alpha, alpha/mu); 

    // noninformative prior for mu 
    target += -log(mu);
    // prior for alpha 
    target += -2 * log(z0 + alpha);
}

/**********************************************************************
 * THE END
 **********************************************************************/