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I have a die with $N$ sides. It has a hidden transitive order which has to be uncovered. At the start of a trial, I throw the die until I get the highest-ranking side. I then throw the die until I get the next highest-ranking side, and repeat until side $1$, the lowest-ranking, remains. After one trial, I therefore have some confidence in a ranking $N > N - 1 > N - 2 >\ \cdots\ > 2 > 1$.

I represent each side as a Beta distribution with a uniform prior, which can be updated as the trial proceeds. I can have as many trials as are necessary to uncover the ranking.

Is there an elegant way to model this?

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This is unclear to me. Do you know the ranking? It sounds as if you don't, but how do you know when to move to the next stage of the trial if you don't? And there doesn't seem to be anything in there that tells you anything about the ranking. Are you told when you get the highest-ranking side? But if so, you'd have complete knowledge of the ranking at the end of just one trial? – joriki Nov 23 '11 at 15:32
Assume N = 6. You don't know the ranking, but you know that 6 was preferred over a subset of {5, 4, 3, 2, 1}. Likewise, you knowledge that 5 was preferred over a subset of {4, 3, 2, 1}. You'd have evidence of a rank after one trial ( 6 > 5 > 4 > 3 > 2 > 1 ), but I was hoping to model the change in confidence over e.g. 100 trials that this is the actual rank, based on some Beta distributions to record the evidence. – Alan Nov 23 '11 at 15:50
I'm afraid that hasn't made it any clearer. Since you say the order is transitive, that one ranking determines it completely. You'll need to explain in what sense you're speaking of "evidence" and "confidence" when you seem to be describing a situation in which all there is to know can be deduced with certainty. – joriki Nov 23 '11 at 22:06
My apologies. I assume an agent that is gathering information about the ranking of the items. It starts with a representation of each item as a Beta distribution (initilalised to Beta(a = 0, b = 0)). After one trial, we can record (a = times selected, b = times seen) and these can be updated to reflect that e.g. 6 > 5 > 4 > 3 > 2 > 1, but assume that as it is only one trial, the agent is not justified in saying this is the rank with 100% certainty. So I am trying to represent this gathering of evidence of rank in the correct way. – Alan Nov 24 '11 at 12:19
There is no information in any of what you've written about how "the agent is not justified in saying this is the rank with 100% certainty". You first describe the process as if it occurs deterministically; then you want to treat it probabilistically; but without knowing anything about the source and extent of the uncertainty, nothing can be said about the update. Clearly the posterior probabilities must depend on the probabilities for somehow observing an incorrect ranking, and since you've said nothing about that, nothing can be said about the posterior probabilities. – joriki Nov 24 '11 at 12:27
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