Would you agree that the information-theoretic increase in the amount of adaptive data in a single organism is still limited by O(1) bits in Mackay’s model?
I can’t really process this query until you relate the words you’ve used to the math MacKay uses, i.e., give me some equations. Also, Eliezer is pretty clearly talking about information in populations, not just single genomes. For example, he wrote, “This 1 bit per generation has to be divided up among all the genetic variants being selected on, for the whole population. It’s not 1 bit per organism per generation, it’s 1 bit per gene pool per generation.”
Eliezer,
I’ve thought hard about your reply, but it’s not clear to me what the distinction is between bits on a hard drive (or in a genome) and information-theoretic bits. One bit on a hard drive answers one yes-or-no question, just like an information-theoretic bit.
The third section of the paper is entitled “The maximum tolerable rate of mutation”. (MacKay left the note “This section needs checking over...” immediately under the title, so there’s room for doubt about his argument.) MacKay derives the rate of change of fitness in his models as a function of mutation rate. He concludes (as you did) that the maximum genome size scales as the inverse of the mutation rate, but only when mutation is the sole source of variation. He makes the claim that maximum genome size scales as the inverse of the square of the mutation rate when crossover is used.
It seems to me that this is a perfect example of your idea that one doesn’t really understand something until the equations are written down. MacKay has tried to do just that. Either his math is wrong, or the idea that truncation can only give on the order of one bit of selection pressure is just the wrong abstraction for the job.
(Just as a follow up, MacKay demonstrates that the key difference between mutation and crossover is that the full progeny (i.e., progeny before truncation) generated by mutation have a smaller average fitness than their parents, while the full progeny generated by crossover have average fitness equal to their parents’.)
Rolf,
I can’t really process this query until you relate the words you’ve used to the math MacKay uses, i.e., give me some equations. Also, Eliezer is pretty clearly talking about information in populations, not just single genomes. For example, he wrote, “This 1 bit per generation has to be divided up among all the genetic variants being selected on, for the whole population. It’s not 1 bit per organism per generation, it’s 1 bit per gene pool per generation.”
Eliezer,
I’ve thought hard about your reply, but it’s not clear to me what the distinction is between bits on a hard drive (or in a genome) and information-theoretic bits. One bit on a hard drive answers one yes-or-no question, just like an information-theoretic bit.
The third section of the paper is entitled “The maximum tolerable rate of mutation”. (MacKay left the note “This section needs checking over...” immediately under the title, so there’s room for doubt about his argument.) MacKay derives the rate of change of fitness in his models as a function of mutation rate. He concludes (as you did) that the maximum genome size scales as the inverse of the mutation rate, but only when mutation is the sole source of variation. He makes the claim that maximum genome size scales as the inverse of the square of the mutation rate when crossover is used.
It seems to me that this is a perfect example of your idea that one doesn’t really understand something until the equations are written down. MacKay has tried to do just that. Either his math is wrong, or the idea that truncation can only give on the order of one bit of selection pressure is just the wrong abstraction for the job.
(Just as a follow up, MacKay demonstrates that the key difference between mutation and crossover is that the full progeny (i.e., progeny before truncation) generated by mutation have a smaller average fitness than their parents, while the full progeny generated by crossover have average fitness equal to their parents’.)