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Is there a better principaled approach to #1 than Monte Carlo sampling from beta distributions?

The Jeffrey posterior in the answer is closed form and Bayesian. The other answer is a profile likelihood.

Neither involve Monte Carlo sampling. Both are general and principled.

The answer looks a bit simplistic compared to the question (as I interpreted it at least). In order to estimate the risk of incorrect ordering you have to calculate P(p2 > p1) where p1 and p2 are drawn from different beta distributions. AFAIK there’s no closed form expression for that probability (so Monte Carlo is one possible approach).

The question doesn’t ask for that —- it explicitly asks us to control for over estimation of the fraction — although I rather like your interpretation as an extension.

Ok. I may have read a bit too much into this paragraph:

> Order the items above, smallest fraction first, whilst taking into account the uncertainty in the number of trials to bound the probability of over-estimating each fraction.

But why mention ordering if you’re not looking for statistical reasoning around the ordering in particular?

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