Dynamics of the Genetic Operators

We will now describe how one can cast mutation and crossover into a simple formal framework. This will be done for the example of nearly the simplest fitness function. 

What is the average effect of the mutation operator for the simple paramagnet The fitness of a mutated member can be written as 

From these single mutation events, the distribution of the population after mutation is obtained by averaging over the parents and mutation events. Let us neglect finite-population effects in the mutation step which are much smaller than those in selection. Then, for all following purposes, the members of the population can be considered as independent and the population average simplifies to 

These first two orders can be expressed in terms of m. Any higher order terms cannot be derived from a knowledge of m alone. However, approximating the 

The first two cumulants of that calculation are equal to (3.23). The third and higher cumulants differ by terms that depend on spin configurations in the population. The spin dependent terms are then chosen to assume their most likely values, given the known fitness distribution. This yields the most accurate estimate for the dynamics on the basis of the fitnesses alone. 

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