# Taking a partial derivative with respect to a model

• Jan 10th 2011, 03:56 PM
jrobot
Taking a partial derivative with respect to a model
"[...] These variables give a measure of correspondence between signal $\displaystyle {\bf X}(n)$ and model $\displaystyle {\bf M}_h$ relative to all other models, $\displaystyle h'$. A mechanism of concept formation and learning, an internal dynamics of the modeling fields (MF) is defined as follows,

$\displaystyle {\bf S}_h = {\bf S}_h + \alpha \displaystyle\sum\limits_{n}f(h|n)\left(\frac{\par tial \ln l(n|h)}{\partial {\bf M}_h}\right)\frac{\partial {\bf M}_h}{\partial {\bf S}_h}$

[...]

Parameter $\displaystyle \alpha$ determines the iteration step and speed of convergence of the MF system; $\displaystyle N_h$ can be interpreted as a number of signals $\displaystyle {\bf X}(n)$ associated with or coming from a concept object $\displaystyle n$. As already mentioned, in the MF internal dynamics, similarity measures are adapted so that their fuzziness is matched to the model uncertainty. [...]"
http://www.leonid-perlovsky.com/4%20-%20Mehler.pdf (page 368)

What does it mean to take a partial derivative with respect to a model $\displaystyle {\bf M}_h$ i.e. $\displaystyle \left(\frac{\partial \ln l(n|h)}{\partial {\bf M}_h}\right)$?

The article says that $\displaystyle l(n\mid h)$ is a contraction for $\displaystyle l(\mathbf{X}(n)\mid\mathbf{M}_h(n))$, i.e., $\displaystyle l$ is a function of two arguments. So apparently $\displaystyle \frac{\partial \ln l(n|h)}{\partial {\bf M}_h}$ is the derivative with respect to the second argument.