Posterior estimates of rate and shape of gamma distribution are dependent

Modeling
3

Just for the information. I computed the off-diagonal element of fisher matrix for Gamma distribution parametrized by log geometric mean and log scale and it is indeed equal to zero suggesting that log geometric mean and log scale provide orthogonal parametrization.

For those interested, here is my STAN implementation of inverse digamma function:

functions{
    vector invdigamma(vector x){
        vector[num_elements(x)] y; vector[num_elements(x)] L;
        for (i in 1:num_elements(x)){
            if (x[i]==digamma(1)){ 
                y[i]=1;
            }else{ if (x[i]>=-2.22){
                y[i]=(exp(x[i])+0.5);
            }else{
                y[i]=1/(x[i]-digamma(1));
        }}}
        L=digamma(y)-x;
        while (min(L)>10^-12){
            y=y-L ./trigamma(y);
            L=digamma(y)-x;
        }
        return y;}
    real invdigammaR(real x){
        real y; real L;
        if (x==digamma(1)){ 
            y=1;
        }else{ if (x>=-2.22){
            y=(exp(x)+0.5);
        }else{
            y=1/(x-digamma(1));
        }}
        L=digamma(y)-x;
        while (L>10^-12){
            y=y-L ./trigamma(y);
            L=digamma(y)-x;
        }
        return y;
    }
}
5 Likes