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The Schema Theorem and Price's Theorem Lee Altenberg

Summary: 1
The Schema Theorem and Price's Theorem
Lee Altenberg
Institute of Statistics and Decision Sciences
Duke University, Durham, NC, USA 27708­0251
Internet: altenber@dynamics.org
Holland's Schema Theorem is widely taken to be the foundation for explanations of the
power of genetic algorithms (GAs). Yet some dissent has been expressed as to its implica­
tions. Here, dissenting arguments are reviewed and elaborated upon, explaining why the
Schema Theorem has no implications for how well a GA is performing. Interpretations of
the Schema Theorem have implicitly assumed that a correlation exists between parent and
offspring fitnesses, and this assumption is made explicit in results based on Price's Covari­
ance and Selection Theorem. Schemata do not play a part in the performance theorems
derived for representations and operators in general. However, schemata re­emerge when
recombination operators are used. Using Geiringer's recombination distribution represen­
tation of recombination operators, a ``missing'' schema theorem is derived which makes
explicit the intuition for when a GA should perform well. Finally, the method of ``adap­
tive landscape'' analysis is examined and counterexamples offered to the commonly used
correlation statistic. Instead, an alternative statistic --- the transmission function in the


Source: Altenberg, Lee - Department of Information and Computer Science, University of Hawai'i at Manoa


Collections: Computer Technologies and Information Sciences