### Partial Derivatives of the CGF.

To obtain the SPA to the PDF of , we need the joint cumulant generating function (CGF) of and its partial derivatives. The joint CGF is defined by

where log is the vector function which operates on each element of its argument and is the vector sum, which adds up all the elements of the argument. If we define as the element-by-element reciprocal of    , and as the diagonal -by- matrix with elements equal to the elements of    , it is straight-forward to show that the gradient vector of is the -by-1 vector

and the -by- Hessian matrix of is

Baggenstoss 2017-05-19