I appreciate your explicitly laying out issues with the Laplace prior! I found this helpful.
The approach to picking a prior here which I feel least uneasy about is something like: “take a simplicity-weighted average over different generating processes for distributions of hinginess over time”. This gives a mixture with some weight on uniform (very simple), some weight on monotonically-increasing and monotonically-decreasing functions (also quite simple), some weight on single-peaked and single-troughed functions (disproportionately with the peak or trough close to one end), and so on…
If we assume a big future and you just told me the number of people in each generation, I think my prior might be something like 20% that the most hingey moment was in the past, 1% that it was in the next 10 centuries, and the rest after that. After I notice that hingeyness is about influence, and causality gives a time asymmetry favouring early times, I think I might update to >50% that it was in the past, and 2% that it would be in the next 10 centuries.
(I might start with some similar prior about when the strongest person lives, but then when I begin to understand something about strength the generating mechanisms which suggest that the strongest people would come early and everything would be diminishing thereafter seem very implausible, so I would update down a lot on that.)
I’m sympathetic to the mixture of simple priors approach and value simplicity a great deal. However, I don’t think that the uniform prior up to an arbitrary end point is the simplest as your comment appears to suggest. e.g. I don’t see how it is simpler than an exponential distribution with an arbitrary mean (which is the max entropy prior over R+ conditional on a finite mean). I’m not sure if there is a max entropy prior over R+ without the finite mean assumption, but 1/x^2 looks right to me for that.
Also, re having a distribution that increases over a fixed time interval giving a peak at the end, I agree that this kind of thing is simple, but note that since we are actually very uncertain over when that interval ends, that peak gets very smeared out. Enough so that I don’t think there is a peak at the end at all when the distribution is denominated in years (rather than centiles through human history or something). That said, it could turn into a peak in the middle, depending on the nature of one’s distribution over durations.
I appreciate your explicitly laying out issues with the Laplace prior! I found this helpful.
The approach to picking a prior here which I feel least uneasy about is something like: “take a simplicity-weighted average over different generating processes for distributions of hinginess over time”. This gives a mixture with some weight on uniform (very simple), some weight on monotonically-increasing and monotonically-decreasing functions (also quite simple), some weight on single-peaked and single-troughed functions (disproportionately with the peak or trough close to one end), and so on…
If we assume a big future and you just told me the number of people in each generation, I think my prior might be something like 20% that the most hingey moment was in the past, 1% that it was in the next 10 centuries, and the rest after that. After I notice that hingeyness is about influence, and causality gives a time asymmetry favouring early times, I think I might update to >50% that it was in the past, and 2% that it would be in the next 10 centuries.
(I might start with some similar prior about when the strongest person lives, but then when I begin to understand something about strength the generating mechanisms which suggest that the strongest people would come early and everything would be diminishing thereafter seem very implausible, so I would update down a lot on that.)
I’m sympathetic to the mixture of simple priors approach and value simplicity a great deal. However, I don’t think that the uniform prior up to an arbitrary end point is the simplest as your comment appears to suggest. e.g. I don’t see how it is simpler than an exponential distribution with an arbitrary mean (which is the max entropy prior over R+ conditional on a finite mean). I’m not sure if there is a max entropy prior over R+ without the finite mean assumption, but 1/x^2 looks right to me for that.
Also, re having a distribution that increases over a fixed time interval giving a peak at the end, I agree that this kind of thing is simple, but note that since we are actually very uncertain over when that interval ends, that peak gets very smeared out. Enough so that I don’t think there is a peak at the end at all when the distribution is denominated in years (rather than centiles through human history or something). That said, it could turn into a peak in the middle, depending on the nature of one’s distribution over durations.