It seems to me that there are (at least) two ways of specifying observers given a physical world-model, and two corresponding ways this would affect anthropics in Solomonoff induction:
You could specify their location in space-time. In this case, what matters isn’t the number of copies, but rather their density in space, because observers being more sparse in the universe means more bits are needed to pin-point their location.
You could specify what this type of observer looks like, run a search for things in the universe matching that description, then pick one off the list. In this case, again what matters is the density of us(observers with the sequence of observations we are trying to predict) among all observers of the same type.
Which of the two methods ends up being the leading contributor to the Solomonoff prior depends on the details of the universe and the type of observer. But either way, I think the Presumptuous Philosopher’s argument ends up being rejected: in the ‘searching’ case, it seems like different physical theories shouldn’t affect the frequency of different people in the universe, and in the ‘location’ case, it seems that any physical theory compatible with local observations shouldn’t be able to affect the density much, because we would perceive any copies that were close enough.
Yeah, the log(n) is only the absolute minimum. If you’re specifying yourself mostly by location, then for there to be n different locations you need at least log(n) bits on average (but in practice more), for example.
But I think it’s plausible that the details can be elided when comparing two very similar theories—if the details of the bridging laws are basically the same and we only care about the difference in complexity, that difference might be about log(n).
Another thing, I don’t think Solomonoff Induction would give an advantage of log(n) to theories with n observers. In the post you mention taking the discrete integral of 2−log(n)=1n to get log scaling, but this seems to be based on the plain Kolmogorov complexity C(n), for which log(n) is approximately an upper bound. Solomonoff induction uses prefix complexity K(n), and the discrete integral of 2−K(n) converges to a constant. This means having more copies in the universe can give you at most a constant advantage.
(Based on reading some other comments it sounds like you might already know this. In any case, it means S.I. is even more anti-PP than implied in the post)
It seems to me that there are (at least) two ways of specifying observers given a physical world-model, and two corresponding ways this would affect anthropics in Solomonoff induction:
You could specify their location in space-time. In this case, what matters isn’t the number of copies, but rather their density in space, because observers being more sparse in the universe means more bits are needed to pin-point their location.
You could specify what this type of observer looks like, run a search for things in the universe matching that description, then pick one off the list. In this case, again what matters is the density of us(observers with the sequence of observations we are trying to predict) among all observers of the same type.
Which of the two methods ends up being the leading contributor to the Solomonoff prior depends on the details of the universe and the type of observer. But either way, I think the Presumptuous Philosopher’s argument ends up being rejected: in the ‘searching’ case, it seems like different physical theories shouldn’t affect the frequency of different people in the universe, and in the ‘location’ case, it seems that any physical theory compatible with local observations shouldn’t be able to affect the density much, because we would perceive any copies that were close enough.
Yeah, the log(n) is only the absolute minimum. If you’re specifying yourself mostly by location, then for there to be n different locations you need at least log(n) bits on average (but in practice more), for example.
But I think it’s plausible that the details can be elided when comparing two very similar theories—if the details of the bridging laws are basically the same and we only care about the difference in complexity, that difference might be about log(n).
Another thing, I don’t think Solomonoff Induction would give an advantage of log(n) to theories with n observers. In the post you mention taking the discrete integral of 2−log(n)=1n to get log scaling, but this seems to be based on the plain Kolmogorov complexity C(n), for which log(n) is approximately an upper bound. Solomonoff induction uses prefix complexity K(n), and the discrete integral of 2−K(n) converges to a constant. This means having more copies in the universe can give you at most a constant advantage.
(Based on reading some other comments it sounds like you might already know this. In any case, it means S.I. is even more anti-PP than implied in the post)
Actually I had forgotten about that :)