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buybuydandavis comments on A probability question - Less Wrong
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You can write that, and it's likely possible in some cases, but step back and think, Does this really make sense to say in the general case?
I just don't think so. The whole problem with mixture of experts, or combining multiple data sources, is that the marginals are not in general independent.
Sure, it's not generically true, but PhilGoetz is thinking about a specific application in which he claims that it is justified to regard the expert estimates as independent (conditional on Q, of course). I don't know enough about the relevant domain to assess his claim, but I'm willing to take him at his word.
I was just responding to your claim that the detectors must suck if the correlation is small. That would be true if the unconditional correlation were small, but its not true if the correlation is small conditional on Q.