One of today’s Bible readings is Matthew 13:24-30, the Parable of the Wheat and Weeds. In the ESV, it goes like this:
He put another parable before them, saying, “The kingdom of heaven may be compared to a man who sowed good seed in his field, but while his men were sleeping, his enemy came and sowed weedsamong the wheat and went away. So when the plants came up and bore grain, then the weeds appeared also. And the servantsof the master of the house came and said to him, ‘Master, did you not sow good seed in your field? How then does it have weeds?’ He said to them, ‘An enemy has done this.’ So the servants said to him, ‘Then do you want us to go and gather them?’ But he said, ‘No, lest in gathering the weeds you root up the wheat along with them. Let both grow together until the harvest, and at harvest time I will tell the reapers, Gather the weeds first and bind them in bundles to be burned, but gather the wheat into my barn.’”
Today I was also reading Cory Doctorow’s new novel Little Brother. In chapter 8, the Department of Homeland Security seeks to root out terrorists by testing an individual persons’ behavior against some normative behavior. The test passes and the person is innocent (wheat); the test fails and the person is a terrorist (weed). Trouble is: statistical error makes this kind of discernment difficult. In the words of Cory:
If you ever decide to do something as stupid as build an automatic terrorism detector, here’s a math lesson you need to learn first. It’s called “the paradox of the false positive,” and it’s a doozy.
Say you have a new disease, called Super-AIDS. Only one in a million people gets Super-AIDS. You develop a test for Super-AIDS that’s 99 percent accurate. I mean, 99 percent of the time, it gives the correct result — true if the subject is infected, and false if the subject is healthy. You give the test to a million people.
One in a million people have Super-AIDS. One in a hundred people that you test will generate a “false positive” — the test will say he has Super-AIDS even though he doesn’t. That’s what “99 percent accurate” means: one percent wrong.
What’s one percent of one million?
1,000,000/100 = 10,000
One in a million people has Super-AIDS. If you test a million random people, you’ll probably only find one case of real Super-AIDS. But your test won’t identify one person as having Super-AIDS. It will identify 10,000 people as having it.
Your 99 percent accurate test will perform with 99.99 percent inaccuracy.
That’s the paradox of the false positive. When you try to find something really rare, your test’s accuracy has to match the rarity of the thing you’re looking for. If you’re trying to point at a single pixel on your screen, a sharp pencil is a good pointer: the pencil-tip is a lot smaller (more accurate) than the pixels. But a pencil-tip is no good at pointing at a single atom in your screen. For that, you need a pointer — a test — that’s one atom wide or less at the tip.
This is the paradox of the false positive, and here’s how it applies to terrorism:
Terrorists are really rare. In a city of twenty million like New York, there might be one or two terrorists. Maybe ten of them at the outside. 10/20,000,000 = 0.00005 percent. One twenty-thousandth of a percent.
That’s pretty rare all right. Now, say you’ve got some software that can sift through all the bank-records, or toll-pass records, or public transit records, or phone-call records in the city and catch terrorists 99 percent of the time.
In a pool of twenty million people, a 99 percent accurate test will identify two hundred thousand people as being terrorists. But only ten of them are terrorists. To catch ten bad guys, you have to haul in and investigate two hundred thousand innocent people.
Guess what? Terrorism tests aren’t anywhere close to 99 percent accurate. More like 60 percent accurate. Even 40 percent accurate, sometimes.
What this all meant was that the Department of Homeland Security had set itself up to fail badly. They were trying to spot incredibly rare events — a person is a terrorist — with inaccurate systems.
Is it any wonder we were able to make such a mess?
Telling the difference between actual wheat and a weed is pretty easy. That test has just about no chance of error. But for someone without a green thumb, telling the difference between the good plants and the weeds isn’t as easy. Just the same way, you can’t tell the difference between a terrorist and an innocent person very easily. Cory estimates that 40-60% accuracy is about right, but I’m not as optimistic.
It’s an interesting collision of two readings today. Saint Matthew and Cory Doctorow both put the same paradox of false positive into the mouths of Jesus and Marcus. Be careful when trying to root out the weeds (you probably aren’t as good at it as you think)! For now, the wheat and the weeds will just have to live together.