/**********************************************************************
 * t-distribution model with known degrees of freedom 
 * introduce a latent variable z 
 * Peng Zeng @ Auburn University 
 * 2025-09-17 
 * 
 * model: 
 *    y ~ t(df, mu, sigma) 
 * equivalent model with a latent vriable z  
 *    y ~ normal(mu, sigma / sqrt(z)) 
 *    z ~ gamma(df/2, df/2) 
 * prior: 
 *    f(mu, sigma) ~ 1 / sigma 
 **********************************************************************/

data {
    int<lower=1> n;                // sample size 
    real y[n];                     // observations 
    real<lower=0> df;              // degrees of freedom 
}
parameters { 
    real mu;                       // mean
    real<lower=0> sigma;           // standard deviation 
    vector<lower=0>[n] z;   
}
model {
    y ~ normal(mu, sigma ./ sqrt(z));   
    z ~ gamma(df/2, df/2); 
    target += -log(sigma);         // noninformative prior 
}

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