kjellstrom's blog
Because the Gaussian distribution is the exponential of squared parameters, the proof of its theorem for adaptation is a rather simple matter, which should be understandable at the high school level. Because the theorem is valid for all Gaussians and all regions of acceptability (even probability functions) it is in principle sufficient to see the proof for a Gaussian with variance = 1 in a single parameter. The proof is easily extended to an arbitrary number of parameters.
Gaussian adaptation – based on the selection of individuals - is closely connected to the Hardy-Weinberg law for its validity as a model of evolution showing its maximization of mean fitness, which can’t be shown by Fisher’s fundamental theorem, 1930. This blog gives a simple proof of the law for individuals, readable on the high school level. Randomness also seems to be an advantage for the survival of the collective.
Would it be possible to make a scientific approach to the question of equal dignity of all human beings? If all religions and -isms are products of evolution, it should perhaps be possible. According to the theory of Gaussian adaptation, evolution strives to solve a collective survival problem by increasing mean fitness and diversity in parallel.
If I understand macro-evolution rightly, it means that some divine force switch from one species to another. Recalling that the development from fertilized egg (a unicellular organism) to adult individual (many-cellular) may be seen as a stepwise modified recapitulation of the evolution of the individual, there is perhaps no need for any macro-evolution, at least not for our own species.
Creationists have reason to doubt the theory based on Fisher’s fundamental theorem of natural selection published in 1930. It relies on the assumption that a gene (allele) may have a fitness of its own being a unit of selection. Historically this way of thinking has also influenced our view of egoism as the most important force in evolution.
Mayr: What Evolution is, 2001, states that “it is sometimes claimed that evolution, by producing order, is in conflict with the ‘law of entropy’ of physics, according to which evolutionary change should produce an increase of disorder. Actually there is no conflict, because the entropy law is valid for closed systems only, whereas the evolution of a species of organisms can reduce entropy at the expense of the environment and the sun supplies a continuing input of energy.”
Creationists are right in the sense that random events do not produce order. But they have produced an enormous amount of disorder (average information)represented by millions of different species and billions of different individuals in certain species, in agreement with the entropy law.
As an example we may mention the definition of fitness given by Maynard Smith, 1998, in the following way: ”Fitness is a property, not of an individual, but of a class of individuals – for example homozygous for allele A at a particular locus. Thus the phrase ’expected number of offspring’ means the average number, not the number produced by some one individual."
It seems to me that this definition denies the fitness of the individual. Nevertheless, the individual fitness is needed, because otherwise the “expected average number of offspring” from a certain class of individuals can’t be determined.
According to Kjellström, 1969, (see reference in the list
http://en.wikipedia.org/wiki/Gaussian_adaptation#References)
a connection between GA and information theory is the average speed of stepwise random walks inside a high dimensional simplex region. It turns out that the speed is asymptotically proportional to (see also the point 7 in blog “Gaussian adaptation as a model of evolution”)
- P log(P),
where P is the probability that a random step will lead to a new feasible position inside the simplex. Maximum speed is obtained when P = 1/e = 0.37.
A plausible interpretation of this is that 1/P is proportional to the time/work needed to find a step leading to a feasible position, while –log(P) is the self-information obtained when such a step may be taken. Thus, – P log(P) may be seen as a measure of efficiency; information divided by the work/time needed to get the information. In addition GA maximizes the average information of a Gaussian distribution
To see a possible connection between Gaussian adaptation, GA, and a brain we may first look at a 2-dimensional computer simulation of the process according to the figure below. It relies on the assumption that neuron kernels may add, synapses may multiply and axons may delay signal values (in accordance with the theory of digital filters).
A pocketful of theorems makes it plausible to use Gaussian adaptation as a simple second order statistical model of the evolution of quantitative traits provided that those traits are Gaussian distributed, or nearly so.
In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.
