# Solving the Likelihood Equation

The principle for solving the moment equation is to choose a value §, simulating graphs x^{(1)}, x^{(2)},..., x^{(M)}, calculating the sample equivalent *z**§ =* M(z(x^{(1)}) + z(x^{(2)}) + ••• + z(x^{(M)})) of *E**§(**z**(**X**))* over this sample, and then checking whether *z**§* is equal to *z**(**x** _{o}b_{s}).* If the difference

*z*

*§ —*

*z*

*(*

*x*

_{o}b_{s}) is not 0, we choose another value

*§*and repeat the process. This is done until we find a value

*§*for which

*z*

*§ —*

*z*

*(*

*x*

_{o}b_{s}) = 0, the MLE. In other words, we gradually change the parameter values to adjust the distributions of statistics. There will always be a sampling variability in

*z*§, the size of which depends on

*M*.

A “brute force” procedure of trying different values of *§* and determining whether *z**§* — *z**(**x*_{o}b_{s}) = 0 is not very effective. Two main approaches for solving *z**§ — **z**(**x*_{o}b_{s}) = 0 have been proposed in the literature - importance sampling and stochastic approximation. The first is the default of statnet, and the second is used in PNet and SIENA, and is available in statnet.