Information Theory, Inference, and Learning ... - MAELabs UCSD

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272 19 — Why have Sex? Information Acquisition and Evolution
and variance
m(1 − m)
G
m
. (19.4)
G
If the population of parents has mean δf(t) and variance σ2 (t) ≡ βm/G, then
the child population, before selection, will have mean (1 − 2m)δf(t) and variance
(1+β)m/G. Natural selection chooses the upper half of this distribution,
so the mean fitness and variance of fitness at the next generation are given by
δf(t+1) = (1 − 2m)δf(t) + α
m
(1 + β) , (19.5)
G
σ 2 (t+1) = γ(1 + β) m
, (19.6)
G
where α is the mean deviation from the mean, measured in standard deviations,
and γ is the factor by which the child distribution’s variance is reduced
by selection. The numbers α and γ are of order 1. For the case of a Gaussian
distribution, α = 2/π 0.8 and γ = (1 − 2/π) 0.36. If we assume that
the variance is in dynamic equilibrium, i.e., σ2 (t+1) σ2 (t), then
γ(1 + β) = β, so (1 + β) = 1
, (19.7)
1 − γ
and the factor α (1 + β) in equation (19.5) is equal to 1, if we take the results
for the Gaussian distribution, an approximation that becomes poorest when
the discreteness of fitness becomes important, i.e., for small m. The rate of
increase of normalized fitness is thus:
df
dt
−2m δf +
which, assuming G(δf) 2 ≫ 1, is maximized for
mopt =
m
, (19.8)
G
1
, (19.9)
16G(δf) 2
at which point,
df 1
= . (19.10)
dt opt 8G(δf)
So the rate of increase of fitness F = fG is at most
dF
dt
= 1
8(δf)
per generation. (19.11)
For a population with low fitness (δf < 0.125), the rate of increase of fitness
may exceed 1 unit per generation. Indeed, if δf 1/ √ G, the rate of increase, if
m = 1/2, is of order √ G; this initial spurt can last only of order √ G generations.
For δf > 0.125, the rate of increase of fitness is smaller than one per generation.
As the fitness approaches G, the optimal mutation rate tends to m = 1/(4G), so
that an average of 1/4 bits are flipped per genotype, and the rate of increase of
fitness is also equal to 1/4; information is gained at a rate of about 0.5 bits per
generation. It takes about 2G generations for the genotypes of all individuals
in the population to attain perfection.
For fixed m, the fitness is given by
δf(t) =
1
2 √ mG (1 − c e−2mt ), (19.12)