I doubt I can easily convince you that the prior I’ve chosen is objectively best, or even that it is better than the one you used. Prior-choice is a bit of an art, rather like choice of axioms. But I hope you see that it does show that the whole thing comes down to whether you choose a prior like you did, or another reasonable alternative… Additionally, if you didn’t know which of these priors to use and used a mixture with mine weighted in to a non-trivial degree, this would also lead to a substantial prior probability of HoH.
I think this point is even stronger, as your early sections suggest. If we treat the priors as hypotheses about the distribution of events in the world, then past data can provide evidence about which one is right, and (the principle of) Will’s prior would have given excessively low credence to humanity’s first million years being the million years when life traveled to the Moon, humanity becoming such a large share of biomass, the first 10,000 years of agriculture leading to the modern world, and so forth. So those data would give us extreme evidence for a less dogmatic prior being correct.
If we treat the priors as hypotheses about the distribution of events in the world, then past data can provide evidence about which one is right, and (the principle of) Will’s prior would have given excessively low credence to humanity’s first million years being the million years when life traveled to the Moon, humanity becoming such a large share of biomass, the first 10,000 years of agriculture leading to the modern world, and so forth.
On the other hand, the kinds of priors Toby suggests would also typically give excessively low credence to these events taking so long. So the data doesn’t seem to provide much active support for the proposed alternative either.
It also seems to me like different kinds of priors are probably warranted for predictions about when a given kind of event will happen for the first time (e.g. the first year in which someone is named Steve) and predictions about when a given property will achieve its maximum value (e.g. the year with the most Steves). It can therefore be consistent to expect the kinds of “firsts” you list to be relatively bunched up near the start of human history, while also expecting relevant “mosts” (such as the most hingey year) to be relatively spread out.
That being said, I find it intuitive that periods with lots of “firsts” should tend to be disproportionately hingey. I think this intuition could be used to construct a model in which early periods are especially likely to be hingey.
I don’t think I agree with this, unless one is able to make a comparative claim about the importance (from a longtermist perspective) of these events relative to future events’ importance—which is exactly what I’m questioning.
I do think that weighting earlier generations more heavily is correct, though; I don’t feel that much turns on whether one construes this as prior choice or an update from one’s prior.
I think this point is even stronger, as your early sections suggest. If we treat the priors as hypotheses about the distribution of events in the world, then past data can provide evidence about which one is right, and (the principle of) Will’s prior would have given excessively low credence to humanity’s first million years being the million years when life traveled to the Moon, humanity becoming such a large share of biomass, the first 10,000 years of agriculture leading to the modern world, and so forth. So those data would give us extreme evidence for a less dogmatic prior being correct.
On the other hand, the kinds of priors Toby suggests would also typically give excessively low credence to these events taking so long. So the data doesn’t seem to provide much active support for the proposed alternative either.
It also seems to me like different kinds of priors are probably warranted for predictions about when a given kind of event will happen for the first time (e.g. the first year in which someone is named Steve) and predictions about when a given property will achieve its maximum value (e.g. the year with the most Steves). It can therefore be consistent to expect the kinds of “firsts” you list to be relatively bunched up near the start of human history, while also expecting relevant “mosts” (such as the most hingey year) to be relatively spread out.
That being said, I find it intuitive that periods with lots of “firsts” should tend to be disproportionately hingey. I think this intuition could be used to construct a model in which early periods are especially likely to be hingey.
I don’t think I agree with this, unless one is able to make a comparative claim about the importance (from a longtermist perspective) of these events relative to future events’ importance—which is exactly what I’m questioning.
I do think that weighting earlier generations more heavily is correct, though; I don’t feel that much turns on whether one construes this as prior choice or an update from one’s prior.
A related outside-view argument for the HoH being more likely to occur in earlier centuries:
New things must happen more frequently in earlier centuries because over time, we will run out of new things to do.
HoH will probably occur due to some significant thing (or things) happening.
HoH must coincide with the first occurrence of this thing, because later occurrences of the same thing or similar things cannot be more important.
If we accept these premises, this justifies using a diminishing prior like Laplace.