I need some help with calculating the derivatives of the logarithm of following density:
$p(X,Y, \lambda_{x},\lambda_{y},\gamma)=exp(-\lambda_{x}-\lambda_{y} - \gamma) \frac{\lambda_{x}^X}{X!} \frac{\lambda_{y}^Y}{Y!} \sum_{k=0}^{min(X,Y)} {X \choose k} {Y \choose k} k! (\frac{\gamma}{\lambda_{x}\lambda_{y}}) ^k$
Taking the log: $log(p(X,Y, \lambda_{x},\lambda_{y},\gamma))=(-\lambda_{x}-\lambda_{y} - \gamma) +log(\frac{\lambda_{x}^X\lambda_{y}^Y}{X!Y!}) + log\left[ \sum_{k=0}^{min(X,Y)} {X \choose k} {Y \choose k} k! (\frac{\gamma}{\lambda_{x}\lambda_{y}}) ^k\right]$
where
$\lambda_{ij}=\left[\begin{array}{ccc}\lambda_{xij}\\\lambda_{yij}\end{array}\right]$
and
$\lambda_{xij}=exp(\alpha_{i}-\beta_{j}+\delta)$
$\lambda_{yij}=exp(\alpha_{j}-\beta_{i})$
$z=( \alpha_{1},..., \alpha_{n}, \beta_{1},....,\beta_{n})$
Question is how do I deal with such derivative: $\frac{ \partial log(p)}{ \partial z}$. Using the chain rule I obtain: $\frac{ \partial log(p)}{ \partial z} = \frac{ \partial \lambda'_{ij}}{ \partial z} \times \frac{ \partial log(p)}{ \partial \lambda_{ij}}$
$\frac{ \partial^2 log(p)}{ \partial z^2} = \frac{ \partial \lambda'_{ij}}{ \partial z} \times \frac{ \partial^2 log(p)}{ \partial \lambda_{ij}^2} \times \frac{ \partial \lambda_{ij}}{ \partial z}+\left( \frac{ \partial^2 \lambda_{ij}}{ \partial z^2} \times \frac{ \partial log(p)}{ \partial \lambda_{ij}}\right)$
How should I multiply those matrices? The dimensions don't seem to match, also I'm not sure if I correctly used the chain rule.