Adding onto this a little, here’s a toy model of viral genetic diversity based on my high-school level biology.
Suppose the virus’ DNA starts out as 000 (instead of ACTG for simplicity), and it needs to mutate into 111 to become stronger. Each individual reproduction event has some small probability p of flipping one of these bits. Some bit flips cause the virus to fail to function altogether, while others have no or negligible effect on the virus. As time goes on, the number of reproduction events starting from a given bitstring grows exponentially, so the likelihood of getting one more 1 grows exponentially as well. However, each time you jump from 000 to 100, it’s not as if all other copies of 000 turn into 100, so making the next jump takes a while of waiting on lots of copies of 100 to happen. And then some 101 appears, and there’s no jump for a while again as that strain populates.
The upshot is that you imagine the viral population to be “filling out the Hamming cube” one bitflip at a time and the weight of each bitstring is the total number of viruses with that code, and a genuinely new strain only appears when all 3 bits get flipped in some copy. But:
(a) The more total copies of the virus there is, the faster a bad mutation happens (speed scaling linearly).
(b) Assuming that some mutations require multiple independent errors to occur (which seems likely?), the virus population is “making incremental research progress” over time by spreading out across the genetic landscape towards different strains, even when no visibly different strains occur.
Given the high dimension of the search space, I think (b) is negligible and the linear model (a) of your first comment is better. In low dimension the boundary of the unit sphere is small and you can have a lot of copies on the inside, having to pass through the sphere to reach new terrain. Whereas, in high dimensions, the population will quickly thin out and all be unique, so what matters is the total volume of space explored, not how long it takes to get anywhere.
I’m not totally convinced this is the right way to think about it, any given useful mutation will depend on some constant number of coordinates flipping, so in this high-dimensional space you’re talking about, useful mutations would look like affine subspaces of low codimension. When you project down to the relevant few dimensions, there’s probably more copies of virus than points to fit in, and it takes a long time for them to spread out.
I guess it depends on the geometry of the problem, whether there are a small number of relevant mutations that make a difference, each with a reasonable chance of being reached, or a huge number of relevant mutations each of which is hard to reach.
Adding onto this a little, here’s a toy model of viral genetic diversity based on my high-school level biology.
Suppose the virus’ DNA starts out as 000 (instead of ACTG for simplicity), and it needs to mutate into 111 to become stronger. Each individual reproduction event has some small probability p of flipping one of these bits. Some bit flips cause the virus to fail to function altogether, while others have no or negligible effect on the virus. As time goes on, the number of reproduction events starting from a given bitstring grows exponentially, so the likelihood of getting one more 1 grows exponentially as well. However, each time you jump from 000 to 100, it’s not as if all other copies of 000 turn into 100, so making the next jump takes a while of waiting on lots of copies of 100 to happen. And then some 101 appears, and there’s no jump for a while again as that strain populates.
The upshot is that you imagine the viral population to be “filling out the Hamming cube” one bitflip at a time and the weight of each bitstring is the total number of viruses with that code, and a genuinely new strain only appears when all 3 bits get flipped in some copy. But:
(a) The more total copies of the virus there is, the faster a bad mutation happens (speed scaling linearly).
(b) Assuming that some mutations require multiple independent errors to occur (which seems likely?), the virus population is “making incremental research progress” over time by spreading out across the genetic landscape towards different strains, even when no visibly different strains occur.
Given the high dimension of the search space, I think (b) is negligible and the linear model (a) of your first comment is better. In low dimension the boundary of the unit sphere is small and you can have a lot of copies on the inside, having to pass through the sphere to reach new terrain. Whereas, in high dimensions, the population will quickly thin out and all be unique, so what matters is the total volume of space explored, not how long it takes to get anywhere.
I’m not totally convinced this is the right way to think about it, any given useful mutation will depend on some constant number of coordinates flipping, so in this high-dimensional space you’re talking about, useful mutations would look like affine subspaces of low codimension. When you project down to the relevant few dimensions, there’s probably more copies of virus than points to fit in, and it takes a long time for them to spread out.
I guess it depends on the geometry of the problem, whether there are a small number of relevant mutations that make a difference, each with a reasonable chance of being reached, or a huge number of relevant mutations each of which is hard to reach.