Partial Derivatives of the CGF.

To obtain the SPA to the PDF of ${\bf z}$, we need the joint cumulant generating function (CGF) $c_z($$\lambda$$)$ of ${\bf z}$ and its partial derivatives. The joint CGF is defined by

$$c_z($$$$\mbox{\boldmath$\lambda$}$$$$)= \log( g_z($$$$\mbox{\boldmath$\lambda$}$$$$)) = - {\rm sum}\left(\mbox{\bf log}({\underline{\bf 1}}- {\bf P} \mbox{\boldmath$\lambda$}) \right)$$

where log is the vector $\log$ function which operates on each element of its argument and ${\rm sum}(\;)$ is the vector sum, which adds up all the elements of the argument. If we define $\phi$$^{-1}($$\lambda$$)$ as the element-by-element reciprocal of ${\underline{\bf 1}}- {\bf P}$   $\lambda$, and $\Phi$$($$\lambda$$)$ as the diagonal $N$-by-$N$ matrix with elements equal to the elements of ${\underline{\bf 1}}- {\bf P}$   $\lambda$, it is straight-forward to show that the gradient vector of $c_z($$\lambda$$)$ is the $M$-by-1 vector

   $$\mbox{\boldmath$\delta$}$$$$($$$$\mbox{\boldmath$\lambda$}$$$$) \stackrel{\mbox{\tiny$\Delta$}}{=} {\partial \over \partial \m... ...) = {\bf P}^\prime \; \mbox{\boldmath$\phi$}^{-1}(\mbox{\boldmath$\lambda$}),$$

and the $M$-by-$M$ Hessian matrix of $c_z($$\lambda$$)$ is

$${\bf C}_z($$$$\mbox{\boldmath$\lambda$}$$$$) \stackrel{\mbox{\tiny$\Delta$}}{=}{\partial^2 \over \partial \... ...P}^\prime \; \mbox{\boldmath$\Phi$}^{-2}(\mbox{\boldmath$\lambda$})\; {\bf P},$$