Sunday, January 3, 2010

Rare Terrorists and Bayes' Rule

Matt Yglesias says "Very Rare Terrorists are Very Hard to Find":
monitoring the UK’s 1.5 million Muslims is a lost cause. If you have a 99.9 percent accurate method of telling whether or not a given British Muslim is a dangerous terrorist, then apply it to all 1.5 million British Muslims, you’re going to find 1,500 dangerous terrorists in the UK. But nobody thinks there are anything like 1,500 dangerous terrorists in the UK. I’d be very surprised if there were as many as 15. And if there are 15, that means you’re 99.9 percent accurate method is going to get you a suspect pool that’s overwhelmingly composed of innocent people.
This is basically Bayes' Rule at work. A hypothetical 1.5 million-person pool of suspects in Great Britain, only 1/10,000 of whom are terrorists. An identification method that is 99.9% effective means that with probability 0.999 a Muslim will be identified as a terrorist when he is one, and a 0.001 probability that he will be identified as a terrorist when s/he is not. Thus, you will have about 1,500 identified as terrorists by the intelligence technique. Thus, the probability of someone is a terrorist given that he
has been identified as such by the method is just 0.01 (although this
is a marked improvement over 0.0001 obtained by profiling all Muslims)!
The good news is that you have almost certainly netted all 15 terrorists in that pool. The bad news is that you've arrested, and harassed 1,485 innocent British Muslims (about 99% of those identified as terrorists).

I've done a similar calculations for using torture and the so-called 1% doctrine. Here is a paper by Hugo Mialon, Sue Mialon and Maxwell Stinchcombe on the subject. They find that legalizing torture creates a disincentive for using other means of investigation, even in cases where there is low evidence of terrorist involvement.
HT: TC at MR

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