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(xobs). If the difference z§ — z(xobs) is not 0, we choose another value § and repeat the process. This is done until we find a value § for which z§ — z(xobs) = 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(xobs) = 0 is not very effective. Two main approaches for solving z§ — z(xobs) = 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.