Why the schema theorem is correct also in the presence of stochastic effects

J. Holland's (1975) schema theorem has been criticised by D.B. Fogel and A. Ghozeil (1997, 1998, 1999) for not being able to correctly estimate the expected proportion of a schema in the population when fitness-proportionate selection is used in the presence of noise or other stochastic effects. This is incorrect for two reasons. Firstly, the theorem in its original form is not applicable to this case. If the quantities involved in schema theorems are random variables, the theorems must be interpreted as conditional statements. Secondly, the conditional versions of Holland and other researchers' schema theorems are indeed very useful to model the sampling of schemata in the presence of stochasticity. In this paper, I show how one can calculate the correct expected proportion of a schema in the presence of stochastic effects when only selection is present, using a conditional interpretation of Holland's schema theorem. In addition, I generalise this result (again using schema theorems) to the case in which crossover, mutation and selection-with-replacement are used. This can be considered as an exact schema theorem that is applicable both in the presence and in the absence of stochastic effects.