It seems like using your logic, we can similarly say that the evidence against Guede screens off the evidence of the bra clasp and knife. Is that correct?
Not really. Unlike the fact of the murder of Knox’s roommate, the bra clasp and knife are essentially independent of the evidence against Guede.
While it is perhaps true, using your labels above, that P(A|E&D) > P(A|E&~D), the difference between these quantities is surely very small compared to the difference between P(A|D) and P(A|E&D).
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
Unlike the fact of the murder of Knox’s roommate, the bra clasp and knife are more or less independent of the evidence against Guede.
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”?
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”?
Not really. Unlike the fact of the murder of Knox’s roommate, the bra clasp and knife are essentially independent of the evidence against Guede.
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).