Tag Archives: Probability

Stereotyping, and Rare but Important Events

Phil Arena has an interesting but problematic piece up at Duck of Minerva, entitled “Bayes, Stereotyping, and Rare Events.” The substantive topic of the post is a recent survey of Muslims that I’m not too interested in. But Phil uses statistics to mask a deeply flawed and irrelevant conclusion:

Put simply, the probability that you’d be mistaken to assume that someone who belongs to group Y is likely to commit or have committed act X simply because most such acts are committed by members of group Y grows exponentially higher as X becomes rarer. The reason you should not assume that a person is a terrorist just because they’re Muslim, then, is not just that this is politically incorrect and likely to offend delicate liberal sensibilities. It’s that it’s almost certainly incorrect, full stop.

The first and last sentences in that paragraph have almost nothing to do with each other. Phil’s conclusion is irrelevant, and the “full stop” leaves the most important part of the conclusion unsaid.

And Phil’s not alone in such a mistake. Take for example an recent statement on the NPR program “Tell Me More” by Fernando Vila. Fernando is responding to a statement that a disproportionate fraction of violent crimes in New York City are committed by African Americans:

VILA: Well, I mean, the notion of paranoia is a good one and Mario’s statistics actually sort of feed into that – into this culture of paranoia. I mean, the vast majority of black people are not committing crimes.

VILA: You know, it’s like to say, I don’t know – the vast majority of hosts on NPR are white males. That doesn’t mean that every time I encounter a white male on the street I assume he’s a host of NPR. You know, it’s just a backwards way of looking at it

Phil and Fernando make exactly the same mistake: false assuming the cost of a “false positive” (accidentally marking someone as suspicious) is the same as the cost of a “false negative” (accidentally marking someone as not suspicious). But the truth is all errors are not equal.

The cost of a mistake is a function of the severity of the mistake.

Is the cost to society of 1 false positive (falsely placing an individual under suspicion of terrorism) the same as the cost to society of 1 false negative (falsely removing suspicion from an actual terrorist)? No, of course not, but Phil’s post is based that on fallacy. Otherwise his conclusion makes no sense.

There is a serious question as to where we should become indifferent to the trade-off — 10:1? 100:1? 1:1000000? — but it is certainly not 1:1.

Likewise, Fernando’s statement on NPR is irrelevant. While the consequence of guessing an individual’s employment status at NPR might be 1:1 (few would care either way), the costs of falsely assuming someone would attack you is far less than the cost of falsely assuming an individual will not attack you. Again, there is a question of trade-offs — 1000:1, 10000:1, 1000000:1? — but the cost of all errors are not identical.

Now, obviously Phil and Fernando had different motives here. Phil’s obviously trying to popularize some basic statistics, while Fernando is doubtless ignorant of basic statistics. But in both cases an unwary audience will be led astray into thinking all errors are equally important.

Prejudice and Bias

Only stupid people judge once.

For the rest of us, the world is a pretty exciting place. There’s always new things to consider, surprising details come up, and the “sure thing” of yesterday becomes the “maybe!” of today (and vice versa!)

At any given time, what we think of a person, a situation, or an event is our judgment. What we thought about it the last time (and the time before that) was our prejudgement, or prejudice. Our prejudices form our bias. What we will think in the future is our Monday morning quarterbacking, the difference between which and what your judgment is our hindsight bias.

If you have the correct prejudices, your hindsight bias will be l0w because your judgment will be correct. The Zimmerman Affair provides a good example of how this could work.

Knowing nothing else about the case, reasonable prejudices would provide a pretty good clue as to who violently attacked whom in the following pair.

person 1:
Sex: Male
Age: Upper 20s
Height: 5’7
Workplace: Desk
Fitness: Out out shape
Ethnicity: Hispanic

person 2:
Sex: Male
Age: Upper teens
Height: 5’11
Workplace: Unemployed (full time school)
Fitness: Athletic
Ethnicity: African-American

With this prejudice, all facts are filtered (this is analogous to the Bayesian process of “updating priors“) and one would come to the same conclusion that the jury in the Zimmerman case did: person 1 is not guilty on all counts.

But of course, only stupid people judge once.

Time goes on, our priors of person 1 are continuously updated , though there is a censorship effect in gathering new information about person 2.

There are remarkably easy ways to predict when some people have the wrong biases. But that is a post for a different time.

Go-Up or Go-Next?

The idea that there are two cultures in academic life, a culture focused on the humanities and another on science, is not a new one. The famous “Two Cultures” lecture is more than fifty years old, and Brother Guy Consolmagno identifies instances of the two cultures in medieval Catholic Europe in his book of adventures.

Jason Lee Steorts, a writer for the National Review Online, defended NRO’s dismissal of John Derbyshire, demonstrates that by criticizing Derbyshire’s controversial article for being hypocritical. In 2012, Derbyshire writes in paragraph 4:

The default principle in everyday personal encounters is, that as a fellow citizen, with the same rights and obligations as yourself, any individual black is entitled to the same courtesies you would extend to a nonblack citizen. That is basic good manners and good citizenship. In some unusual circumstances, however—e.g., paragraph (10h) below—this default principle should be overridden by considerations of personal safety.

While two years previously, in a speech on race relations, he said

Group differences are statistical truths. They exist in an abstract realm quite far removed from our everyday personal experience. They tell you nothing about the person you just met.

This would be hypocracy, unless you believe the fundemental principles of statistics have undergone a revolution in the past two years. Which, of course, they have.

There are two ways of understanding statistics. The terms “Frequentist” and “Probabilistic” are thrown around here, but to me those words are more confusing than helpful. So I will call them the go-up and go-next views of statistics.

The Go-Up view of statistics is that statistics measures the population from which an observation comes from. The appropriate way to go-up is to wait until you have a sufficient number of observations. and then generalize about the population from our observations. This is the method that Derbyshire was describing in 2010. A large number of observations of academic performance show consistent gaps between black and white learners. Because we’re “going-up” from observations to populations, we can conclude some things about the population, and how outcomes in the population should work-out over all, but it makes no sense to try to predict any given student’s success based on this. We’re going-up, not going-next.

The Go-Next view of statistics is that statistics gives us the likelihood of something being true, based on what has come before. In Go-Next statistics, population-averages are besides the point. What matters is guessing what’s going to happen, next, based on what you’ve seen before. The whole point is to guess what’s going to work for individuals you know only a few things about, based on your experience with other individuals who shared some things with the new strangers.

Both the Go-Up and Go-Next interpretations of statistics are hundreds of years old. Go-Up statisitcs strikes many as more beautiful. Go-Next as, perhaps, more practical, more commercial, more technical. Astronomers use go-up statistics. Weathermen use go-next statistics.

The Internet changed everything.

Academics pay attention to reality. Professors, like most people, respond to the incentives of power, influence, and money. Companies like Google, Facebook, Apple, and my employer do not care much about abstract ideas like “What can we infer about internet users in general based on the observations we collect.” Instead, they care, very, very deeply, about making you delighted. Because people will spend money to be delighted.

When you log onto your Facebook screen, or type a search into Google, or click the genius buttons in iTunes, you want it to just work. You want the perfect update, the perfect site, the perfect song. Advertisers want the perfect ad for you.

In this context, the view of statistics that Derbyshire outlined in 2010:

Group differences are statistical truths. They exist in an abstract realm quite far removed from our everyday personal experience. They tell you nothing about the person you just met.

Is just stupid. Facebook doesn’t care about the group differences between men and women. It cares that when you log in, it can give you an update from your favorite sports team, or gossip from your favorite celebrity, or whatever. Never before in history has so much math been used to make you happy.

It’s all about you.

It’s all about guessing, based on what has come before, what’s best for you.

It’s all about guessing, based on prior observations, who you are, what you will do, and what you will like.

These major companies have been hiring those with statistical literacy very heavily for more than a decade. Professors, who seek, money, fame, and power, know what these large potential sources of money, fame, and power want, and teach their products — their students –accordingly.

The superstructure of science changes as the infrastructure of the economy changes. The Go-Next philosophy of statistics, once the peasant stepchild of the serene Go-Up interpretation, now reigns supreme.

The unfolding victory of Go-Next Statistics matters much, much more than, say, the Copernican Revolution. The number of people whose daily conversations were actually impacted by Copernicus may have been a few dozen, all involved in the Papal-Academic complex.

How many times a day does Facebook’s decision of which news to share impact you?

How many times a day does Google’s decision of what sites to show impact you?

How many times a day does your iPod’s decision of what music to play impact you?

Now, back to Derbyshire.

Mr. Derbyshire was born in 1945. His training is in Go-Up statistics. It took a complete revolution in statistics to change his view of it. That view clearly changed in the last 2 years.

We’ve all lived thru the revolution of Go-Next statistics. Derbyshire realizes it. Steorts, clearly, does not.

There are two cultures of knowledge, the humanities and the sciences. Part of Derbyshire’s intention of writing “The Talk – Nonblack Version” appears to have been to highlight this. If so, I think he succeeded.