Mathematics In Gangland

Scott Greenfield, font of so much that I can riff off of, has a complaint about gang experts who try to paint every action by the defendant as related to his membership in a gang:

If a defendant has a tattoo, the expert will testify that tattoos are “brands” typically worn by gang members.  If the tattoo happens to say “Tiffany”, then the testimony is changed ever so slightly to accommodate, by the expert then saying that gang members typically brand themselves with the names of their girlfriends.  You get the message.  No matter what the evidence, the defendant can’t win.  It’s always connectible to being a gang member, according to the expert.

Aside from the implication by Scott that police gang experts are pulling answers out of their ass don’t really know much about gangs, I can also see something of a logical problem with the so-called expert’s theory. The cop’s statement that gang members have specific types of tattoos is a logical statement: If he’s a gang member, then he’ll have a specific tattoo, say of a snarling dog. If we try to get all mathematical, the statement it would look like this:

gang member ==> dog tattoo

But that’s not what the prosecutor wants the jury to believe. The prosecutor doesn’t care about the tattoo. He’s trying to prove that the defendant is a gang member. He’s trying to prove the converse:

dog tattoo ==> gang member

The problem is, as a matter of math, the truth of a statement does not imply the truth of its converse. Even if it’s true that all gang members have dog tattoos, it doesn’t mean all people with dog tattoos are gang members. It’s easier to see with a more obvious example: All gang members have noses:

gang member ==> nose

but that doesn’t in any way prove

nose ==> gang member

That is, not all people with noses are gang members.

So why is it that “all people with noses are gang members” is obviously wrong, but “all people with dog tattoos are gang members” seems plausible? The answer is a little more complicated because it involves the real world, the statistics of experimental design for testing hypotheses, and the availability heuristic.

Suppose you wanted to test the hypothesis “all people with X are gang members,” where X is either “noses” or “dog tattoos.” You’d do it by setting up an experiment—in this case a survey of the population—to look for counterexamples to the hypothesis. That is, you’d try to find people who have X but are not gang members. Finding even one proves it’s not absolute truth.

The real world is a little fuzzy—especially in the social sciences—and our methods of testing are less than perfect, so real-world hypothesis testing usually involves testing a statistical relationship. In this case, we’d be testing “people with X have a high probability of being gang members” and we’d still be looking for people who have X but are not gang members. The more such counterexamples we find, the lower the probability of a relationship.

A scientific test of this kind of hypothesis would involve conducting random surveys and gathering enough data to reach statistically reliable conclusions. But when it’s not a scientific investigation, when it’s just us trying to figure something out, we don’t do a scientific study. We just try to think of counterexamples.

When X is “noses” it’s easy. We all know lots of people with noses, and nearly all of them are not gang members. Such a large number of counterexamples makes it easy to destroy the hypothesis that all people with noses are gang members.

When the hypothesis is “all people with dog tattoos are gang members,” it’s a little harder to think of counterexamples, simply because dog tattoos are so rare that we may not know of anybody who has one. Our inability to find counterexamples makes the hypothesis seem plausible.

This way of thinking is called the availability heuristic. We assume something is likely because we can easily bring to mind examples. The more examples we can think of, the more likely we believe it to be.

Although the availability heuristic is not as rigorous and generalized as conducting a scientific investigation, in essence it’s a similar process. A scientific investigation gathers data using randomized trials, controlled studies, and careful surveys and then analyzes the data to arrive at results. The availability heuristic does the same kind of analysis, but it works only on the data we have in our heads at that moment. There is no data gathering process.

The availability heuristic is a perfectly valid way of thinking about our day-to-day world, about which we have lots of data but don’t have time to gather more. It tends to fail us, however, when thinking about parts of the world with which we are unfamiliar. That’s why we have science.

What are the practical implications of all this philosophy when it comes to thinking about gang experts? Probably not much. But if I ever find myself on a jury listening to this kind of testimony, I hope I’ll keep a few points in mind.

Basically, any assertion of a general rule—all fish have fins, all cats have fur, all people with dog tattoos are gang members—is equivalent to an assertion that counterexamples do not exist: There are no fish that don’t have fins, there are no cats that don’t have fur, there are no people who have dog tattoos who are not in gangs.

(Actually, the gang expert will likely invoke several indicators of gang membership in combination—gang tattoos, gang hats, gang shoe laces, gang jewelry—and the standard in the courtroom is not that there are absolutely no counterexamples, but rather that counterexamples are rare enough that they do not constitute a cause for reasonable doubt. Nevertheless, an assertion of a general rule is still an assertion about the rarity of counterexamples.)

So if you hear someone say that dog tattoos are sign of gang membership, you should be wondering why that person believes there are no (or few) counterexamples. Remember, it’s not just about gangs. It’s also about tattoos. If he’s really an expert on gangs, he may very well have observed that gang members have dog tattoos, but how does he know that non-gang-members don’t have dog tattoos as well? He’d have to know a lot about tattoo prevalence in society at large. In addition to being a gang expert, he’d also have to be a tattoo expert.

Or at least he’d have to have received reliable information from a tattoo expert or be aware of a scientific study of some kind that addressed the issue. If I were on the jury, I’d want to hear about that.