Decided to provide my solution since others have done so as well.
Solution
The public dataset is approximately symmetrical, so it is very likely that the distribution of the Bernoulli rate is also symmetrical (probability at p is equal to probability at 1-p). Let the probabilities of getting k Rs over all 5 trials for k=0...5 be (a,b,c,c,b,a). Then, from the public dataset, we have a+b/5≈0.252854,4b/5+2c/5≈0.166231,6c/5≈0.161832. These have standard deviation ≈0.0004, which is negligible, so we can treat these as linear equations. Solving, we get a=0.224781,b=0.140359,c=0.13486, and we can then solve for the marginal frequencies b/5a+b/5=0.111020,2c/54b/5+2c/5=0.324512, etc.
Not sure if this (experiment set?) is a good test of priors, since I got an exact answer without having to consider priors, other than the data being symmetrical. (This also means that any symmetric distribution for the Bernoulli rate will result in the same answer.) Though @DaemonicSigil has a similar solution without using symmetry, instead using
maximum entropy as a prior (if i understand it correctly).
Still, almost all reasonable priors will result in very similar outcomes, differing by a factor probably on the order of the standard deviation (around 10−3.) This is likely less than, or at least comparable to, the noise in the actual marginal frequencies.
You’re mostly right. The other solves have given pretty much identical distributions.
Some of your distributions are worse than other distributions. If I run 100,000,000 experiments and calculate the frequencies, some of you will be more off at the fourth decimal point.
The market doesn’t have that kind of precision, and even if it did, I wouldn’t change the resolution criterion. But I can still score you guys myself later on.
I do agree that I should have given much fewer public experiments. Then it would be a better test on priors.
Answer:
[0.111020, 0.324512, 0.5, 0.675488, 0.888980]
I will provide my solution when the market is resolved.
Decided to provide my solution since others have done so as well.
Solution
The public dataset is approximately symmetrical, so it is very likely that the distribution of the Bernoulli rate is also symmetrical (probability at p is equal to probability at 1-p). Let the probabilities of getting k Rs over all 5 trials for k=0...5 be (a,b,c,c,b,a). Then, from the public dataset, we have a+b/5≈0.252854,4b/5+2c/5≈0.166231,6c/5≈0.161832. These have standard deviation ≈0.0004, which is negligible, so we can treat these as linear equations. Solving, we get a=0.224781,b=0.140359,c=0.13486, and we can then solve for the marginal frequencies b/5a+b/5=0.111020,2c/54b/5+2c/5=0.324512, etc.
Not sure if this (experiment set?) is a good test of priors, since I got an exact answer without having to consider priors, other than the data being symmetrical. (This also means that any symmetric distribution for the Bernoulli rate will result in the same answer.) Though @DaemonicSigil has a similar solution without using symmetry, instead using
maximum entropy as a prior (if i understand it correctly).
Still, almost all reasonable priors will result in very similar outcomes, differing by a factor probably on the order of the standard deviation (around 10−3.) This is likely less than, or at least comparable to, the noise in the actual marginal frequencies.
You’re mostly right. The other solves have given pretty much identical distributions.
Some of your distributions are worse than other distributions. If I run 100,000,000 experiments and calculate the frequencies, some of you will be more off at the fourth decimal point.
The market doesn’t have that kind of precision, and even if it did, I wouldn’t change the resolution criterion. But I can still score you guys myself later on.
I do agree that I should have given much fewer public experiments. Then it would be a better test on priors.