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DanielLC comments on Error margins - Less Wrong Discussion
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If you have a probability of probabilities, you can just collapse it into one probability. Suppose you're 50% sure that A has 80% probability, and 50% sure it has 60% probability. Let B be that A has 80% probability.
P(A|B) = 0.8
P(A|!B) = 0.6
P(B) = 0.5
P(A&B) = P(A|B)P(B) = 0.8*0.5 = 0.4
P(A&!B) = P(A|!B)P(!B) = 0.6*0.5 = 0.3
P(A) = P(A&B)+P(A&!B) = 0.4+0.3 = 0.7
So you can just say that A has 70% probability and be done with it. No need for a confidence interval.