Very interesting! I like this formalization/categorization.
Hm… I’d have filed “Why the tails come apart” under “Extremal Goodhart”: this image from that post is almost exactly what I was picturing while reading your abstract example for Extremal Goodhart. Is Extremal “just” a special case of Regressional, where that ellipse is a circle? Or am I missing something?
Regressional: But the best basketball player in the world (according to the NBA MVP award) is just 6′3″ (1.91m), and a randomly selected 7 foot (2.13m) tall person in his 20s would probably be pretty good at basketball but not NBA caliber. That’s regression to the mean; the tails come apart.
Extremal: The tallest person on record, Robert Wadlow, was 8′11″ (2.72m). He grew to that height because of a pituitary disorder, he would have struggled to play basketball because he “required leg braces to walk and had little feeling in his legs and feet”, and he died at age 22. His basketball ability was well below what one might naively predict based on his height and the regression line, and that is unsurprising because the human body wasn’t designed for such an extreme height.
Adversarial: A college basketball player who wants to get drafted early and signed to a big contract grows his hair up, so that NBA teams will measure him as being taller (up to the top of his hair).
Many N.B.A. hopefuls exaggerate their height while in high school or college to make themselves more appealing to coaches and scouts who prefer taller players. Collins, for example, remembers the exact day he picked to experience a growth spurt.
“Media day, my junior year,” Collins, a Stanford graduate, said. “I told our sports information guy that I wanted to be 7 feet, and it’s been 7 feet ever since.”
And:
Victor Dolan, head of the chiropractic division at Doctors’ Hospital in Staten Island, said players could increase their height by being measured early in the morning, because vertebrae become compressed as the day progresses. A little upside-down stretching does not hurt, either.
“If you get measured on an inversion machine, and do it when you first wake up, maybe you could squeeze out an extra inch and a half,” Dolan said.
I’m trying to think of a causal goodheart one. A bad one I came up with is that if someone thinks the reason taller people get better careers is because the hiring committe likes tall people, and so the person wears heels in their shoes, then this is a causal godheart because they’re trying to win on a proxy but in a way causally unrelated to the goal of having a good career.
But everyone knows the true causal story and don’t make this mistake, so it’s not a good example. Is there a causal story people don’t know about? Like perhaps some false belief about winning streaks (as opposed to the standard Kahneman story of regression to the mean).
Causal: An early 1900s college basketball team gets all of their players high-heeled shoes, because tallness causes people to be better at basketball. Instead, the players are slowed and get more foot injuries.
Adversarial: The New York Knicks’ coach, while studying the history of basketball, finds the story about the college team with high heels. He gets marketers to go to other league teams and convince them to wear high heels. A few weeks later, half of the star players in the league are out, and the Knicks easily win the championship.
I thought of almost this exact thing (with stilts). I like it and it is what I plan on using for when I want a simple example. It wish it was more realistic though.
Extremal is not a special case of regressional, but you cannot seperate them completely because regressional is always there. I think the tails come apart is in the right place. (but I didn’t reread the post when I made this)
If you sample a bunch of points from a multivariate normal without the large circular boundary in my example, the points will roughly form an ellipse, and the tails come apart thing will still happen. This would be Regerssional Goodhart. When you add the circular boundary, something weird happens where now you optimization is not just failing to find the best point, but actively working against you. If you optimize weakly for the proxy, you will get a large true value, but when you optimize very strongly you will end up with a low true value.
Very interesting! I like this formalization/categorization.
Hm… I’d have filed “Why the tails come apart” under “Extremal Goodhart”: this image from that post is almost exactly what I was picturing while reading your abstract example for Extremal Goodhart. Is Extremal “just” a special case of Regressional, where that ellipse is a circle? Or am I missing something?
Height is correlated with basketball ability.
Regressional: But the best basketball player in the world (according to the NBA MVP award) is just 6′3″ (1.91m), and a randomly selected 7 foot (2.13m) tall person in his 20s would probably be pretty good at basketball but not NBA caliber. That’s regression to the mean; the tails come apart.
Extremal: The tallest person on record, Robert Wadlow, was 8′11″ (2.72m). He grew to that height because of a pituitary disorder, he would have struggled to play basketball because he “required leg braces to walk and had little feeling in his legs and feet”, and he died at age 22. His basketball ability was well below what one might naively predict based on his height and the regression line, and that is unsurprising because the human body wasn’t designed for such an extreme height.
Great example!
It would be really nice if we had an example like this that worked well for all four types.
Adversarial: A college basketball player who wants to get drafted early and signed to a big contract grows his hair up, so that NBA teams will measure him as being taller (up to the top of his hair).
And:
-- http://www.nytimes.com/2003/06/15/sports/basketball/tall-tales-in-nba-dont-fool-players.html
I’m trying to think of a causal goodheart one. A bad one I came up with is that if someone thinks the reason taller people get better careers is because the hiring committe likes tall people, and so the person wears heels in their shoes, then this is a causal godheart because they’re trying to win on a proxy but in a way causally unrelated to the goal of having a good career.
But everyone knows the true causal story and don’t make this mistake, so it’s not a good example. Is there a causal story people don’t know about? Like perhaps some false belief about winning streaks (as opposed to the standard Kahneman story of regression to the mean).
Causal: An early 1900s college basketball team gets all of their players high-heeled shoes, because tallness causes people to be better at basketball. Instead, the players are slowed and get more foot injuries.
Adversarial: The New York Knicks’ coach, while studying the history of basketball, finds the story about the college team with high heels. He gets marketers to go to other league teams and convince them to wear high heels. A few weeks later, half of the star players in the league are out, and the Knicks easily win the championship.
I thought of almost this exact thing (with stilts). I like it and it is what I plan on using for when I want a simple example. It wish it was more realistic though.
Extremal is not a special case of regressional, but you cannot seperate them completely because regressional is always there. I think the tails come apart is in the right place. (but I didn’t reread the post when I made this)
If you sample a bunch of points from a multivariate normal without the large circular boundary in my example, the points will roughly form an ellipse, and the tails come apart thing will still happen. This would be Regerssional Goodhart. When you add the circular boundary, something weird happens where now you optimization is not just failing to find the best point, but actively working against you. If you optimize weakly for the proxy, you will get a large true value, but when you optimize very strongly you will end up with a low true value.