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AlexSchell comments on How do I make sample size matter? - Less Wrong
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Implicitly, they have an infinite sample size, because the distribution on P(B|A_1) is infinitely precise. Suppose we also wanted to learn P(B|A_1) from the history of the factory: then we might model it as having a fixed rate of defective outputs, and the probability we assign to particular defect rates is a beta distribution. We might start off with a Jeffreys prior and then update as we see the tool produce defective or normal products, eventually ending up with, say, a beta(5.5,95.5) for tool A_1.
Exercise for the reader: given that hyperparameter distribution for P(B|A_1) (and similar ones for A_2 and A_3), do we need the full hyperparameter distribution for all three tools to determine the probability that a known defective output came off of A_1, or can we get the same answer using only a handful of moments from each distribution?
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Yep.
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