### The Equation for Response to Selection and Its Use for Prediction

The Breeder Genetic Algorithm (BGA) was designed according to the theories and methods used in the science of livestock breeding. The prediction of a breeding experiment is based on the response to selection (RS) equation. This equation relates the change in a population's fitness to the standard deviation of its fitness, as well as to the parameters selection intensity and realized heritability. In this paper the exact RS equation is derived for proportionate selection given an infinite population in linkage equilibrium. In linkage equilibrium the genotype frequencies are the product of the univariate marginal frequencies. The equation contains Fisher's fundamental theorem of natural selection as an approximation. The theorem shows that the response is approximately equal to the quotient of a quantity called additive genetic variance, VA, and the average fitness. We compare Mendelian two-parent recombination with gene-pool recombination, which belongs to a special class of genetic algorithms that we call univariate marginal distribution (UMD) algorithms. UMD algorithms keep the genotypes in linkage equilibrium. For UMD algorithms, an exact RS equation is proven that can be used for long-term prediction. Empirical and theoretical evidence is provided that indicates that Mendelian two-parent recombination is also mainly exploiting the additive genetic variance. We compute an exact RS equation for binary tournament selection. It shows that the two classical methods for estimating realized heritabilitythe regression heritability and the heritability in the narrow sensemay give poor estimates. Furthermore, realized heritability for binary tournament selection can be very different from that of proportionate selection. The paper ends with a short survey about methods that extend standard genetic algorithms and UMD algorithms by detecting interacting variables in nonlinear fitness functions and using this information to sample new points.

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