Edit: This comment is wrong and I’m now very embarrassed by it. It was based on a misunderstanding of what the NunoSempere is doing that would have been resolved by a more careful read of the first sentence of the forum post!
Thank you for the link to the timeless version, that is nice!
But I don’t agree with your argument that this issue is moot in practice. I think you should repeat your R analysis with months instead of years, and see how your predicted percentiles change. I predict they will all be precisely 12 times smaller (willing to bet a small amount on this).
This follows from dimensional analysis. How does the R script know what a year is? Only because you picked a year as your trial. If you repeat your analysis using a month as a trial attempt, your predicted mean proof time will then be X months instead of X years (i.e. 12 times smaller).
The same goes for any other dimensionful quantity you’ve computed, like the percentiles.
You could try to apply the linked timeless version instead, although I think you’d find you run into insurmountable regularization problems around t=0, for exactly the same reasons. You can’t get something dimensionful out of something dimensionless. The analysis doesn’t know what a second is. The timeless version works when applied retrospectively, but it won’t work predicting forward from scratch like you’re trying to do here, unless you use some kind of prior to set a time-scale.
Consider a conjecture first made twenty years ago.
If I look at a year as the trial period:
n=20, probability predicted by Laplace of being solved in the next year = 1/(n+2) = 1⁄22 ~= 4.5%
If I look at a month at the trial period:
n = 20 * 12, probability predicted by Laplace of being solved in the next year = the probability that it isn’t solved in any of twelve months = 1 - (1-1/(n+2))^12 = 4.8%
Apologies, I misunderstood a fundamental aspect of what you’re doing! For some reason in my head you’d picked a set of conjectures which had just been posited this year, and were seeing how Laplace’s rule of succession would perform when using it to extrapolate forward with no historical input.
I don’t know where I got this wrong impression from, because you state very clearly what you’re doing in the first sentence of your post. I should have read it more carefully before making the bold claims in my last comment. I actually even had a go at stating the terms of the bet I suggested before quickly realising what I’d missed and retracting. But if you want to hold me to it you can (I might be interpreting the forum wrong but I think you can still see the deleted comment?)
I’m not embarrassed by my original concern about the dimensions, but your original reply addressed them nicely and I can see it likely doesn’t make a huge difference here whether you take a year or a month, at least as long as the conjecture was posited a good number of years ago (in the limit that “trial period”/”time since posited” goes to zero, you presumably recover the timeless result you referenced).
New EA forum suggestion: you should be able to disagree with your own comments.
I model a calendar year as a trial attempt. See here (and the first comment in that post) for a timeless version.
I think that this issue ends up being moot in practice. If we think in terms of something other than years, Laplace would give:
1−(1−1/(d⋅n+2))d
where if e.g., we are thinking in terms of months, d=12
instead of
1/(n+2)
But if we look at the Taylor expansion for the first expression, we notice that its constant factor is
1/(n+2)
and in practice, I think that the further terms are going to be pretty small when n reasonably large.
Alternatively, you can notice that
(1−1/(d⋅n))d converges to the n-th root of e, and that it does so fairly quickly.
Edit: This comment is wrong and I’m now very embarrassed by it. It was based on a misunderstanding of what the NunoSempere is doing that would have been resolved by a more careful read of the first sentence of the forum post!
Thank you for the link to the timeless version, that is nice!
But I don’t agree with your argument that this issue is moot in practice. I think you should repeat your R analysis with months instead of years, and see how your predicted percentiles change. I predict they will all be precisely 12 times smaller (willing to bet a small amount on this).
This follows from dimensional analysis. How does the R script know what a year is? Only because you picked a year as your trial. If you repeat your analysis using a month as a trial attempt, your predicted mean proof time will then be X months instead of X years (i.e. 12 times smaller).
The same goes for any other dimensionful quantity you’ve computed, like the percentiles.
You could try to apply the linked timeless version instead, although I think you’d find you run into insurmountable regularization problems around t=0, for exactly the same reasons. You can’t get something dimensionful out of something dimensionless. The analysis doesn’t know what a second is. The timeless version works when applied retrospectively, but it won’t work predicting forward from scratch like you’re trying to do here, unless you use some kind of prior to set a time-scale.
Consider a conjecture first made twenty years ago.
If I look at a year as the trial period:
n=20, probability predicted by Laplace of being solved in the next year = 1/(n+2) = 1⁄22 ~= 4.5%
If I look at a month at the trial period:
n = 20 * 12, probability predicted by Laplace of being solved in the next year = the probability that it isn’t solved in any of twelve months = 1 - (1-1/(n+2))^12 = 4.8%
As mentioned, both are pretty similar.
Apologies, I misunderstood a fundamental aspect of what you’re doing! For some reason in my head you’d picked a set of conjectures which had just been posited this year, and were seeing how Laplace’s rule of succession would perform when using it to extrapolate forward with no historical input.
I don’t know where I got this wrong impression from, because you state very clearly what you’re doing in the first sentence of your post. I should have read it more carefully before making the bold claims in my last comment. I actually even had a go at stating the terms of the bet I suggested before quickly realising what I’d missed and retracting. But if you want to hold me to it you can (I might be interpreting the forum wrong but I think you can still see the deleted comment?)
I’m not embarrassed by my original concern about the dimensions, but your original reply addressed them nicely and I can see it likely doesn’t make a huge difference here whether you take a year or a month, at least as long as the conjecture was posited a good number of years ago (in the limit that “trial period”/”time since posited” goes to zero, you presumably recover the timeless result you referenced).
New EA forum suggestion: you should be able to disagree with your own comments.
Hey, I’m not in the habit of turning down free money, so feel free to make a small donation to https://www.every.org/quantifieduncertainty
Sure, will do!