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Wei_Dai comments on Experiment: Knox case debate with Rolf Nelson - Less Wrong Discussion
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What you said above was:
Is this the criteria you would use for "screened off" in general? If so, suppose we replace D with F="some DNA evidence exists linking Knox to murder". (E still being "evidence against Guede".) Don't we still have P(A|E&F) - P(A|E&~F) << P(A|F) - P(A|E&F)? To illustrate, P(A|F) = 0.1, P(A|E&F) = 0.01, P(A|E&~F) < 0.001. (These are semi-plausible numbers for illustrating this point, not my actual probabilities.)
In this later comment you say
This seems to make more sense, but I'm still having trouble translating it into a technical definition of "screened off". Can you suggest one?
It's easy to break an approximative definition by applying it to a situation where distinctions between orders of error are important. So any such definition, strictly speaking, has to be considered a sort of analogy or metaphor that may not always be applicable to every context.
Strictly speaking, as you know, "E screens F off from A" means P(A|E&F) = P(A|E&~F). So it seems reasonable to say "E approximately screens F off from A" if |P(A|E&F) - P(A|E&~F)| is small. However, what "small" means is context-dependent. When, above, I declined to apply this terminology to E and F, it was because I was mentally comparing |P(A|E&F) - P(A|E&~F)| to |P(A|E) - P(A|E&F)|, rather than to |P(A|F) - P(A|E&F)|. The latter, of course, is much larger. So I don't suppose I can really stop you from applying the approximative definition of "screening off" in this situation if what you're interested in is P(A|F) vs P(A|E&F) (a large downward jump) rather than P(A|E) vs P(A|E&F) (a small upward jump).
What do you say we table this discussion about "approximately screens off"? (I'm thinking of writing a discussion post asking LW what a good, i.e., generally useful, definition of it would be. Maybe it doesn't have to be context-dependent, or could be less context-dependent, if we talk about P(A|E&F) / P(A|E&~F) instead of P(A|E&F) - P(A|E&~F).)
For now, perhaps you can just tell me what mathematical statement you actually had in mind, when you said "Screened off by the evidence against Rudy Guede"?
P(A|E&D) is much closer to P(A) than to P(A|D).