Iām confused about the methodology here. Laplaceās law of succession seems dimensionless. How do you get something with units of āyearsā out of it? Couldnāt you just as easily have looked at the probability of the conjecture being proven on a given day, or month, or martian year, and come up with a different distribution?
Iām also confused about what this experiment will tell us about the utility of Laplaceās law outside of the realm of mathematical conjectures. If you used the same logic to estimate human life expectancy, for example, it would clearly be very wrong. If Laplaceās rule has a hope of being useful, it seems it would only be after taking some kind of average performance over a variety of different domains. I donāt think its usefulness in one particular domain should tell us very much.
Edit: This comment is wrong and Iām 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 donāt 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 youāve computed, like the percentiles.
You could try to apply the linked timeless version instead, although I think youād find you run into insurmountable regularization problems around t=0, for exactly the same reasons. You canāt get something dimensionful out of something dimensionless. The analysis doesnāt know what a second is. The timeless version works when applied retrospectively, but it wonāt work predicting forward from scratch like youāre 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/ā(n+2) = 1ā22 ~= 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 isnāt solved in any of twelve months = 1 - (1-1/ā(n+2))^12 = 4.8%
Apologies, I misunderstood a fundamental aspect of what youāre doing! For some reason in my head youād picked a set of conjectures which had just been posited this year, and were seeing how Laplaceās rule of succession would perform when using it to extrapolate forward with no historical input.
I donāt know where I got this wrong impression from, because you state very clearly what youāre 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 Iād 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?)
Iām not embarrassed by my original concern about the dimensions, but your original reply addressed them nicely and I can see it likely doesnāt 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 ātrial periodā/āātime since positedā 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.
Iām confused about the methodology here. Laplaceās law of succession seems dimensionless. How do you get something with units of āyearsā out of it? Couldnāt you just as easily have looked at the probability of the conjecture being proven on a given day, or month, or martian year, and come up with a different distribution?
Iām also confused about what this experiment will tell us about the utility of Laplaceās law outside of the realm of mathematical conjectures. If you used the same logic to estimate human life expectancy, for example, it would clearly be very wrong. If Laplaceās rule has a hope of being useful, it seems it would only be after taking some kind of average performance over a variety of different domains. I donāt think its usefulness in one particular domain should tell us very much.
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:
1ā(1ā1/(dā 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
(1ā1/(dā n))d converges to the n-th root of e, and that it does so fairly quickly.
Edit: This comment is wrong and Iām 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 donāt 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 youāve computed, like the percentiles.
You could try to apply the linked timeless version instead, although I think youād find you run into insurmountable regularization problems around t=0, for exactly the same reasons. You canāt get something dimensionful out of something dimensionless. The analysis doesnāt know what a second is. The timeless version works when applied retrospectively, but it wonāt work predicting forward from scratch like youāre 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/ā(n+2) = 1ā22 ~= 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 isnāt solved in any of twelve months = 1 - (1-1/ā(n+2))^12 = 4.8%
As mentioned, both are pretty similar.
Apologies, I misunderstood a fundamental aspect of what youāre doing! For some reason in my head youād picked a set of conjectures which had just been posited this year, and were seeing how Laplaceās rule of succession would perform when using it to extrapolate forward with no historical input.
I donāt know where I got this wrong impression from, because you state very clearly what youāre 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 Iād 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?)
Iām not embarrassed by my original concern about the dimensions, but your original reply addressed them nicely and I can see it likely doesnāt 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 ātrial periodā/āātime since positedā 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, Iām not in the habit of turning down free money, so feel free to make a small donation to https://āāwww.every.org/āāquantifieduncertainty
Sure, will do!