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MrHen comments on Open Thread: February 2010 - Less Wrong
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p(A|X) = p(X|A)*p(A) / ( p(X|A)*p(A) + p(X|~A)*p(~A) )
A = box has diamond
X = box beeped
p(A) = .01
p(X|A) = .88
p(X|~A) = .08
p(A|X) = .88 * .01 / ( .88 * .01 + .08 * .99)
p(A|X) = .0088 / (.0088 + .0792)
p(A|X) = .0088 / .088
p(A|X) = .1
This is different than the likelihood ratio:
LR = p(X|A) / p(X|~A)
LR = .88 / .08
LR = 11
The likelihood ratio can be worded as, "It is 11 times more likely to be a diamond when it beeps." The original formula answers the question, "What is the probability that this beep means a diamond?"
In other words, the likelihood ratio is starting with the contents of a box and asking whether that box is going to beep. p(A|X) is starting with a beep and trying to figure out what that beep means about the contents of the box.
Yes, thanks to Cyan's reply, I've already figured that "it is 11 times more likely to be a diamond when it beeps" than that the beep indicates a false positive. Your reply makes it even more obvious. My whole problem was my ignorance regarding the meaning of the likelihood ratio of testing a random box opposed to the overall probability of a beep. Or in other words, I was unaware that there were actually two different questions being estimated.
Thanks everybody!
Yeah. I was confused at first so I looked it up and wrote it down to make sure I remembered it. Glad it helped. :)