regressionbayesianpoissonstangamma

Why specifiy "exp()" when using log-link in STAN?

I am using STAN to use a gamma-poisson regression to account for over-dispersion in a dataset, in an assignment. I have looked at the solutions to how to make the model in STAN, which is shown below:

model{
    vector[N] lambda;
    scale ~ cauchy( 0 , 1 );
    bf ~ normal( 0 , 1 );
    a ~ normal( 0 , 10 );
    for ( i in 1:N ) {
        lambda[i] = a + bf * fmnnty[i];
        lambda[i] = exp(lambda[i]);
    }
    deaths ~ neg_binomial_2( lambda , scale );
}

The solution specify to use the log-link function to limit the parameter lambda to positive values. This makes sense. However, I don't understand why the STAN-code uses exp() as a link-function. Why is it that the exp() is appropriate to specify a log-link function?

Solution

  • The log-link is employed by exponentiating the lambda. The 'log' component means that you are modeling the log of the mean value (mu):

    log(mu) = lambda

    Which you can rewrite as:

    mu = exp(lambda)