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rstarkov comments on Bayesianism in the face of unknowns - Less Wrong
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For the record, this is not permitted.
It's easy to say this but I don't think this works when you start doing the maths to get actual numbers out. Additionally, if you really take ALL possible distributions then you're already in trouble, because some of them are pretty weird - e.g. the Cauchy distribution doesn't have a mean or a variance.
I'd love to know if there are established formal approaches to this. The only parts of statistics that I'm familiar with assume known distributions and work from there. Anyone?
If I'm not mistaken, the Cauchy distribution wouldn't be included because it's not supported on a bounded interval.
You should probably look at Jaynes's book "Probability Theory: the Language of Science". In particular, I think that the discussion there dealing with the Widget Problem and with Laplace's Rule of Succession may be relevant to your question.
And just as it gets really interesting, that chapter ends. There is no solution provided for stage 4 :/
Odd. The printed book has another page and a half for that chapter, including the solution to Stage 4. (No surprises in the solution - same as stage 3 except you start with 40 fewer Green widgets.)