I think √N is a natural choice for the amount by which the intensity is increased and the response is decreased as the mean (or mode?) of a prior distribution, since we use the same factor increase/decrease for each. But this relies on a very speculative symmetry.
I think deriving √N from the geometric mean between 1 and N is not the best approch, even assuming 1 and N are the “true” minimum and maximum scaling factors. The geometric mean between two quantiles whose sum is 1 (e.g. 0 and 1) corresponds to the median of a loguniform/lognormal distribution, but what we arguably care about is the mean, which is larger.
I’d claim our prior distribution should be somewhat concentrated around √N and its log roughly symmetric around it, so the EV is plausibly close to √N, but it could be higher if the distribution is not concentrated enough, which is also very plausible.
Actually, the difference between the mean and median is much smaller than I expected. For 1/N = 221 M / 86 G = 0.00256 (ratio between the number of neurons of a red junglefowl and human taken from here), the mean and median of a distribution whose 1st and 99th percentiles are 1/N and 1 are:
Lognormal distribution (“very concentrated”): 0.1 and 0.05, i.e. the mean is only 2 times as large as the median.
Loguniform (“not concentrated”): 0.2 and 0.05, i.e. the mean is only 3 times as large as the median.
The mean moral weight of poultry birds relative to humans of 2 I estimated here is 10 times as large as the one respecting the loguniform distribution just above. This makes me think 2 is not an unreasonably high estimate, especially having in mind that there are factors such as clock speed of consciousness which might increase the moral weight of poultry birds relative to humans, instead of decreasing it as the number of neurons.
I think deriving √N from the geometric mean between 1 and N is not the best approch, even assuming 1 and N are the “true” minimum and maximum scaling factors. The geometric mean between two quantiles whose sum is 1 (e.g. 0 and 1) corresponds to the median of a loguniform/lognormal distribution, but what we arguably care about is the mean, which is larger.
I’d claim our prior distribution should be somewhat concentrated around √N and its log roughly symmetric around it, so the EV is plausibly close to √N, but it could be higher if the distribution is not concentrated enough, which is also very plausible.
That makes sense.
Actually, the difference between the mean and median is much smaller than I expected. For 1/N = 221 M / 86 G = 0.00256 (ratio between the number of neurons of a red junglefowl and human taken from here), the mean and median of a distribution whose 1st and 99th percentiles are 1/N and 1 are:
Lognormal distribution (“very concentrated”): 0.1 and 0.05, i.e. the mean is only 2 times as large as the median.
Loguniform (“not concentrated”): 0.2 and 0.05, i.e. the mean is only 3 times as large as the median.
The mean moral weight of poultry birds relative to humans of 2 I estimated here is 10 times as large as the one respecting the loguniform distribution just above. This makes me think 2 is not an unreasonably high estimate, especially having in mind that there are factors such as clock speed of consciousness which might increase the moral weight of poultry birds relative to humans, instead of decreasing it as the number of neurons.