Dr. Mark Humphrys

School of Computing. Dublin City University.

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Picking the fittest

Do only the top n, or the top n percent, get to reproduce?

We could do that, or we could set up a scheme where fitter individuals were more probable to reproduce, and less fit less probable, as follows.

Probability of reproducing - using sum of fitnesses

Let the probability of an individual reproducing be:
p = my fitness
    sum of fitnesses of all creatures

If fitness is positive, then p is a number between 0 and 1, and the sum of all these probabilities is 1. A separate page shows how to make a choice according to probabilities.

One problem with the above is if there is a large population. Let there be one individual with fitness 1, and 100 individuals with fitness 0.2. Then the probability of the fit individual being chosen is only 1/21. One way around this is the following:

Boltzmann "soft max" distribution

Probability of reproducing - using Boltzmann "soft max"

The Boltzmann or "soft max" distribution gives us a flexible way of choosing who reproduces.

Let individual i have fitness f(i). Then its probability of reproducing, p(i), is:

By varying the temperature T, we can change it from only the very fittest reproduce (perhaps towards the end of run) to almost everybody has a chance, probabilities are almost equal (start of run). To be precise, T large at start of run, small towards end of run. The fittest will always be strictly more probable.

Variable population size

Fitter individuals reproduce. If we allow the number of children they have to vary, we may get a variable population size.

Q. As a programmer, what could happen if we have a variable population size?

Fixed population size

Simpler to have fixed population size.

repeat (population size / 2) times 
  pick individual from whole population according to probabilities
  pick individual from whole population according to probabilities
   (may be the same individual!)
  construct 2 new individuals from these 2 parents 

entire old population is now replaced by the new population
 (keep constant population size)

If we do this, make the population size even.

Simple summation of fitness v. Boltzmann

Consider again using simple summation of fitness when deciding who reproduces:
p = my fitness
    sum of fitnesses of all creatures

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