William Morse wrote:
> an588{at}freenet.carleton.ca (Catherine Woodgold) wrote in
> news:coebn4$1m2i$1{at}darwin.ediacara.org: 
> 
> 
>>phillip smith (deletethis-phills{at}ihug.co.nz) writes:
>>
>>>I should have said I am working on a replacement for fitness( see
>>>reply to JE). I think fitness has fatal errors. We keep using it
>>>because we have had no choice. Not that I am sure I an do better.
>>
>>Here is a definition of the fitness of a given gene in context.
>>For this definition to make sense, I assume a multiple-universe
>>model:  that due to quantum-mechanical fluctuations, a given universe
>>at a given point in time develops into multiple future versions
>>(possbilities), over which a meaningful probability metric
>>can be defined.
>>
>>Consider a gene A in a particular individual X in a particular
>>environment.  Consider also all the copies of gene A, for
>>example in siblings, cousins, and other members of the population.
>>
>>Consider the universe U containing individual X.  Consider also
>>the fictional universe U-prime which is exactly like U except that
>>individual X does not exist.
>>
>>Go forward in time 5 generations, and count up
>>the average number of copies of gene A in the population, averaged
>>over the alternate universes with a probabilistic weighting.
>>
>>The fitness of gene A in individual X is the average number
>>of copies of A in the futures of universe U, minus the average
>>number of copies of A in the futures of universe U-prime.
>>Descendents of cousins etc. are also included in the count.
> 
> 
> That's a great definition - up to 5 generations. What happens if all the 
> descendants at 6 generations die without issue? Is the fitness 0, or do 
> we have an arbitrary cutooff at 4 generations? 
> 
One could, of course, define the cut-off as the number of generations 
between the one where the gene acts, and the last relative to be 
affected by the gene's effect.

> So what we would like to have is a definition that, like yours, takes 
> into account alternate future environments, but that can be extended 
> farther into the future. I have a rather fuzzy mathematical intuition 
> that if you somehow combine Markov chains with a discount factor for 
> expected future gains you might come up with a reasonable approximation 
> of a fitness function - recognising that fitness is always going to be 
> ultimately only definable in retrospect.
> 
If you know how the discount declines with generation, this will work as 
well.  Especially as you don't need the whole Markov chain: you only 
need to take expectations.  This simplifies the calculations (at the 
cost of a bit more maths).

The approach is similar to Fisher's original definition of fitness (in 
his "Genetical Theory of NS"), but extended over several generations. 
Until I get involved in an actual application, I can't be bothered to 
work out the precise derivation (or check to see who has already done it).

Bob

-- 
Bob O'Hara
Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax:  +358-9-191 51400
WWW:  http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: www.jnr-eeb.org
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