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@mariopasquato I propose settling it based on the consensus current value of the Hubble parameter. If it turns out cosmology is totally wrong and the universe isn't expanding, H=0 and CMBR wins. If the universe is expanding but not homogeneous enough to define a reasonable value for H.... then I don't care, I'll take that loss.
It would be good to pin down values to define the threshold precisely. I propose the most recent combination values from Adam Riess and the Planck collaboration respectively as of market creation.
@LivInTheLookingGlass I propose using the numbers from 2203.06142:
supernovae: Riess et al. (SH0ES 2022) 2112.04510: 73.04 ± 1.04
CMBR: Aghanim et al. (Planck 2018) 1807.06209: 67.27 ± 0.60
If we average the means, we get 70.155 km/s/Mpc.
If we go for equal stdevs from each mean, we get 69.381 km/s/Mpc, which is 3.52 sigma from both results.
It's pretty tempting to compromise on 70 km/s/Mpc, which is 2.92 sigma below SH0ES and 4.55 sigma above Planck.
I'll leave it up to you though.
@AMS Ok, so when eventually a consensus is reached on H0 you would resolve based on that value of H0 being > 70 or not?
@mariopasquato Actually, I had a better idea (also independently suggested by a friend when I asked for an outside opinion).
We should assume a Gaussian distribution for both measurements, which is not exactly correct but very close. Whichever measurement has the higher log-likelihood at the true value of H0 wins.
This means that:
H0 > 69.44 resolves to NO
59.34 < H0 < 69.44 resolves to YES
H0 < 59.34 resolves to NO
69.44 is 3.616 sigma above Planck and 3.461 sigma below SH0ES.
59.34 is 13.21 sigma below Planck and 13.17 sigma below SH0ES.
Rationale: SH0ES has wider error bars, so it is less wildly incorrect than Planck for small values of H0, even though it predicts a higher mean value. This isn't captured correctly by "equal sigma" because the peak likelihood density is higher for Planck.
Also, more vaguely speaking, this feels right to me because the SH0ES story is "cosmologists are missing something important" and the Planck story is "we basically know what's going on".
Log-likelihood is in general a very good scoring rule with nice properties, although "higher log-likelihood than the other guy" as a binary criterion might not be as good.
I'm not the market creator, just some guy with an opinion.
Note that H0 < 60 is wildly unlikely according to everyone, so this is an edge case anyway. The real question is where to put the threshold between them, since 3.5sigma errors do happen. I think I stand by 69.44 km/s/Mpc.
@AMS Since there will always be some uncertainty in any measurement, this doesn't totally answer the question of how to resolve it if everyone eventually agrees on e.g. a measurement that gives 70 +- 2 km/s/Mpc.
The principled thing is to take the expected log-likelihood of each and compare them, but since we need to know when to resolve, I think it might make more sense to say something like "when there's scientific consensus around a measurement whose 2-sigma error bars exclude 69.44". Or 5 sigma, but that might be a very very long wait.
@AMS Not only 3.5 sigma errors can happen but it may be the case that the error bars are underestimated, possibly in both measurements as well as in the putative future consensus one. Without further information the choice of the threshold determines the outcome
@mariopasquato I think I agree with this take. I am totally willing to be persuaded here, but it seems to me that this is going to be more subjective than just comparing H0. Eventually, I expect that both methods will be able to estimate H0 using different analysis methods than are used today. Whichever method/answer changes the most as a result of new science is then the one that's most wrong
@AMS by "Hubble parameter" you mean the present-time one, so that if the supernovae results are correct about the recent rate of expansion and the CMB results are correct about the ancient rate of expansion this would resolve to NO?
