Venus Mousetrap
Posts: 201
Joined: Aug. 2007
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| Quote (djmullen @ Oct. 11 2007,06:53) | I've got a question about the No Free Lunch theorems. I hope somebody can clarify things for me.
I don't claim to understand the math behind NFL, but from reading the commentary, I get the impression that the theory says that there is no single search strategy that will work perfectly for ALL possible search spaces and data arrangements. For example:
If you have an array set up like this:
1: apple
2: cow
3: echo
4: FTK
5: Geronimo
6: salivate
A simple binary search type algorithm will let you find the word "echo" in the array in three steps or less because the entries are in alphabetical order.
But if the array is arranged like this:
1: cow
2: salivate
3: FTK
4: apple
5: Geronimo
6: echo
The binary search algorithm won't work.
Furthermore, the NFL theorem seems to say that, averaged over all possible arrangements of the data in the array, no single algorithm will work better than a random search, where you choose a number at random, look at the word in that position, see if it's the word you're looking for and repeat the search if it isn't.
Finally, the big thing about the NFL theorem is that it was mathematically proven a few years ago. Is this more or less right? |
My understanding of the critiques of Dembski's work is as follows:
The mathematics is correct - it is true that, averaged over all possible arrangements, no fitness function is greater than any other. But this is not an interesting result. It's like saying that if you stick a sausage in a blender, you get a mushy paste - whereas biologists are more interested in sausages.
It's basically the mathematical equivalent of quote mining - Dembski takes this uninteresting result and tries to claim that it applies to biology, where actually, the situation is quite different. In evolution, we have fitness landscapes, which are full of hills and valleys and mount improbables - in the NFL, we have a random spiky mush.
The result of this is that we can define neighbourhoods in fitness space. One of the examples given by (Haeggstroem?) is that of DNA; we know that, quite often, changing just one nucleotide does not change much at all; the resulting organism has similar or the same fitness. This implicitly defines a neighbourhood; the neighbours to any DNA sequence are all those that can be reached by mutating one nucleotide - and this is how evolution works! (In fact, it's just one of several ways, which gives the fitness neighbourhood even greater population).
But the NFL does not work on this. The NFL is only interested in spiky mushes, where no neighbourhoods can be defined. It would be like a rabbit giving birth to cloud of meat every time one of its nucleotides changed.
So Dembski is math-mining. He goes on, however, to claim that, IF the fitness landscape is so well defined, well, what are the odds of that! It must have been designed! and at that point, you have a man who is quite divorced from reality, and who seems to be claiming that fitness gradients cannot occur in nature. In other words, if you hold Dembski to his words, all the peppered moths have a random chance of being eaten.
Well, that's my understanding of the papers. I am not a mathematician, however, and am more than welcome to be corrected. Mostly it seems that Dembski is claiming stuff so incredibly dumb, underneath the cover of math, that no one will believe he could be so dumb, and instead are left with something incredible.
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