XiXiDu comments on Open Thread: February 2010 - Less Wrong

1 Post author: wedrifid 01 February 2010 06:09AM

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Comment author: XiXiDu 02 February 2010 06:25:36PM *  2 points [-]

Anyone willing to give some uneducated fool a little math coaching? I'm really just starting with math and I probably shouldn't already get into this stuff before reading up more, but it's really bothering me. I came across this page today: http://wiki.lesswrong.com/wiki/Prior_odds

My question, how do you get a likelihood ratio of 11:1 in favor of a diamond? I'm getting this: .88/(.88+7.92)=.1 thus 10% probability for a beep to be a box containing a diamond? Since the diamond-detector is 88% likely to beep on that 1 box and 8% likely to beep on the 99 boxes containing no diamonds. So you have 7.92 false beeps and .88 positive ones which add up to 8.8 beeps of which only .88 are actually boxes containing a diamond?

As of today I'm still struggling with basic algebra. So that might explain my confusion. Though at some point I'll arrive at probability. But I'd be really grateful if somebody could enlighten me now.

Thanks!

Comment author: MrHen 02 February 2010 07:20:01PM *  4 points [-]

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.

Comment author: XiXiDu 02 February 2010 08:22:43PM 0 points [-]

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!

Comment author: MrHen 02 February 2010 08:26:10PM 0 points [-]

Yeah. I was confused at first so I looked it up and wrote it down to make sure I remembered it. Glad it helped. :)

Comment author: ciphergoth 02 February 2010 06:36:58PM 3 points [-]

If you haven't read Bayes Theorem yet, it's definitely the place to start.

Comment author: XiXiDu 02 February 2010 07:55:13PM 0 points [-]

Thanks, yes. It has been on my ToDo-list for some time now. Thought I'd first get more into the basics. I might just make it a priority now.

Comment author: Cyan 02 February 2010 06:34:50PM *  2 points [-]

The likelihood ratio is Pr(beep | diamond) / Pr(beep | empty) = 0.88/0.08 = 11. I was going to say you ought to read the link for "likelihood ratio", but there's nothing there, so you should try the other wiki.

Also, don't think of running the detector over every box; think of testing one box at random.

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