- Let m(H,t) be the number of instances of
H at time t,
û(H,t)
be the observed average fitness of H at time
t, and E(m(H,t+1)) be
the expected number of instances of H at time
t+1.
- Ignoring the effects of crossover and mutation,
`E(m(H,t+1)) = sum_{x in H} f(x)/(bar{f}(t)) = (hat{u}(H,t)) / (bar{f}(t)) m(H,t)`
- Lower bound on the probability that H will survive
single-point crossover:
`S_c(H) >= 1 - p_c ((d(H))/(l-1))`
- Probability that H will survive under mutation of an
instance of H:
`S_m(H) = (1-p_m)^{o(H)}`
- Taking crossover and mutation into account, we have the
Schema Theorem:
`E(m(H,t+1)) >= (hat{u}(H,t)) / (bar{f}(t)) m(H,t) [1 - p_c ((d(H))/(l-1))] (1-p_m)^{o(H)}`
`(E(m(H,t+1)))/(m(H,t)) = (hat{u}(H,t)) / (bar{f}(t)) [1 - p_c ((d(H))/(l-1))] (1-p_m)^{o(H)}`
- Short, lower-order schemas whose average fitness remains above
the mean (building blocks) will receive exponentially increasing numbers
of samples over time.
