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Mathematical secrets of ancient tablet unlocked after nearly a century of study

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Dating from 1,000 years before Pythagoras's theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers

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 Popular Algorithm Is No Better at Predicting Crimes Than Random People

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 Popular Algorithm Is No Better at Predicting Crimes Than Random People

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The COMPAS tool is widely used to assess a defendant's risk of committing more crimes, but a new study puts its usefulness into perspective.

In February 2013, Eric Loomis was found driving a car that had been used in a shooting. He was arrested, and pleaded guilty to eluding an officer. In determining his sentence, a judge looked not just to his criminal record, but also to a score assigned by a tool called COMPAS.

Developed by a private company called Equivant (formerly Northpointe), COMPAS-or the Correctional Offender Management Profiling for Alternative Sanctions-purports to predict a defendant's risk of committing another crime. It works through a proprietary algorithm that considers some of the answers to a 137-item questionnaire.

COMPAS is one of several such risk-assessment algorithms being used around the country to predict hot spots of violent crime, determine the types of supervision that inmates might need, or-as in Loomis's case-provide information that might be useful in sentencing. COMPAS classified him as high-risk of re-offending, and Loomis was sentenced to six years.

He appealed the ruling on the grounds that the judge, in considering the outcome of an algorithm whose inner workings were secretive and could not be examined, violated due process. The appeal went up to the Wisconsin Supreme Court, who ruled against Loomis, noting that the sentence would have been the same had COMPAS never been consulted. Their ruling, however, urged caution and skepticism in the algorithm's use.

Caution is indeed warranted, according to Julia Dressel and Hany Farid from Dartmouth College. In a new study, they have shown that COMPAS is no better at predicting an individual's risk of recidivism than random volunteers recruited from the internet.

"Imagine you're a judge and your court has purchased this software; the people behind it say they have big data and algorithms, and their software says the defendant is high-risk," says Farid. "Now imagine I said: Hey, I asked 20 random people online if this person will recidivate and they said yes. How would you weight those two pieces of data? I bet you'd weight them differently. But what we've shown should give the courts some pause." (A spokesperson from Equivant declined a request for an interview.)

COMPAS has attracted controversy before. In 2016, the technology reporter Julia Angwin and colleagues at ProPublica analyzed COMPAS assessments for more than 7,000 arrestees in Broward County, Florida, and published an investigation claiming that the algorithm was biased against African Americans. The problems, they said, lay in the algorithm's mistakes. "Blacks are almost twice as likely as whites to be labeled a higher risk but not actually re-offend," the team wrote. And COMPAS "makes the opposite mistake among whites: They are much more likely than blacks to be labeled lower-risk but go on to commit other crimes."

Northpointe questioned ProPublica's analysis, as did various academics. They noted, among other rebuttals, that the program correctly predicted recidivism in both white and black defendants at similar rates. For any given score on COMPAS's 10-point scale, white and black people are just as likely to re-offend as each other. Others have noted that this debate hinges on one's definition of fairness, and that it's mathematically impossible to satisfy the standards set by both Northpointe and ProPublica-a story at The Washington Post clearly explains why.

The debate continues, but when Dressel read about it, she realized that it masked a different problem. "There was this underlying assumption in the conversation that the algorithm's predictions were inherently better than human ones," she says, "but I couldn't find any research proving that." So she and Farid did their own.

They recruited 400 volunteers through a crowdsourcing site. Each person saw short descriptions of defendants from ProPublica's investigation, highlighting seven pieces of information. Based on that, they had to guess if the defendant would commit another crime within two years.

On average, they got the right answer 63 percent of their time, and the group's accuracy rose to 67 percent if their answers were pooled. COMPAS, by contrast, has an accuracy of 65 percent. It's barely better than individual guessers, and no better than a crowd. "These are nonexperts, responding to an online survey with a fraction of the amount of information that the software has," says Farid. "So what exactly is software like COMPAS doing?"

Only Equivant can say, and they're not revealing the secrets of their algorithm. So the duo developed their own algorithm, and made it as simple as possible -"the kind of thing you teach undergrads in a machine-learning course," says Farid. They found that this training-wheels algorithm could perform just as well as COMPAS, with an accuracy of 67 percent, even when using just two pieces of data-a defendant's age, and their number of previous convictions. "If you are young and have a lot of prior convictions, you are high-risk," says Farid. "It's kind of obvious."

Other teams have found similar results. Last year, a team of researchers led by Cynthia Rudin from Duke University showed that a basic set of rules based on a person's age, sex, and prior convictions-essentially, an algorithm so simple you could write it on a business card-could predict recidivism as well as COMPAS.

The problem isn't necessarily that COMPAS is unsophisticated, says Farid, but that it has hit a ceiling in sophistication. When he and Dressel designed more complicated algorithms, they never improved on the bare-bones version that used just age and prior convictions. "It suggests not that the algorithms aren't sophisticated enough, but that there's no signal," he says. Maybe this is just as good as it gets. Maybe the whole concept of predicting recidivism is going to stall at odds that are not that much better than a coin toss.

Sharad Goel, from Stanford University, sees it a little differently. He notes that judges in the real world have access to far more information than the volunteers in Dressel and Farid's study, including witness testimonies, statements from attorneys, and more. Paradoxically, that informational overload can lead to worse results by allowing human biases to kick in. Simple sets of rules can often lead to better risk assessments -something that Goel found in his own work. Hence the reasonable accuracy of Dressel and Farid's volunteers, based on just seven pieces of information.

"That finding should not be interpreted as meaning that risk-assessment tools add no value," says Goel. Instead, the message is "when you tell people to focus on the right things, even nonexperts can compete with machine-learning algorithms."

Equivant make a similar point in a response to Dressel and Farid's study, published on Wednesday. "The findings of 'virtually equal predictive accuracy' in this study," the statement says, "instead of being a criticism of the COMPAS assessment, actually adds to a growing number of independent studies that have confirmed that COMPAS achieves good predictability and matches the increasingly accepted AUC standard of 0.70 for well-designed risk assessment tools used in criminal justice."

There have been several studies showing that algorithms can be used to positive effect in the criminal-justice system. "We're not saying you shouldn't use them," says Farid. "We're saying you should understand them. You shouldn't need people like us to say: This doesn't work. You should have to prove that something works before hinging people's lives on it."

"Before we even get to fairness, we need to make sure that these tools are accurate to begin with," adds Dressel. "If not, then they're not fair to anyone."

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A Popular Algorithm Is No Better at Predicting Crimes Than Random People

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The COMPAS tool is widely used to assess a defendant's risk of committing more crimes, but a new study puts its usefulness into perspective. In February 2013, Eric Loomis was found driving a car that had been used in a shooting. He was arrested, and pleaded guilty to eluding an officer.

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What is a number? Everyone who takes high school math learns about the real numbers. These are our old friends on the number line. You can hardly do classical algebra or geometry without them. We use the real numbers so often we find them comfortable and familiar. After all, they are just numbers you write as a (possibly infinite) decimal. They may be long to write, but numbers like

1.41421356237309504880168872420969807856967187537694807317667973799...

don't worry us.

Let's back up a step or two. The integers (that is, the counting numbers and their negatives: 0,1,-1,2,-2,3,-3,,...) aren't very controversial [1]. We can all agree what it means to have 7 goats, no goats, or that I have -3 goats when I owe my neighbor 3 goats. The rational numbers aren't too bad, either. After all, to have 7/5 of a hamburger is to slice two burgers into 5 equal pieces each and to take seven of the pieces. And, again, to have -8/3 of a burger is to owe someone two burgers plus 2/3 of a third [2].

The integers and rationals are down to earth, as numbers go. However, it doesn't take very long before you realize you need more numbers. One day in geometry class you draw a one-by-one square, notice you can draw a straight line which connects the opposite corners, and that line self-evidently has a length. Whatever that length is, it is an honest-to-goodness number (call it D) which exists in nature. At some point, you notice two one-by-one squares can be cut along the diagonal and reassembled into a single, larger square with side length D. On the one hand, the area of this new square is D². On the other hand, it is the area of the two smaller squares taken together. That is, D²=2. The Pythagoreans already knew 2600 years ago there is no rational number whose square equals two. The apocryphal story is the existence of non-rational numbers was a closely held secret for the Pythagoreans, worthy of murder. Nowadays we tell it to school children. So much for the innocence of youth.

We tell children not to worry, this simply means if you draw the number line and mark all the rational numbers, then there are still gaps here and there, like √2, which need to be filled in. If you write it as a decimal, you see √2 is approximately 1.41 and so is just to the left of 1.5 on the number line. Now it's not exactly 1.41 since that would be 141/100, and we know √2 isn't rational. Indeed, for this reason, if we write it as a decimal it must go on forever. Someone tells us the real numbers are those you can write as a decimal, possible infinite. This includes -3, 1.234, and √2. At some point we learn that the rational numbers are exactly the numbers who, when written as a decimal, either stop, or start repeating over and over.

Not very convincingly, we are told we now have all the numbers we need. But, you say, √2 snuck up on us unexpectedly, what if that should happen again? What about the solutions to x²+1=0? E verything is fine! No questions!

Like Martin Luther, you can't help but ask uncomfortable questions. If real numbers are those you can write as a decimal, then you can write easily write down a real number such as

a=.999999999999999...

If it is a number, and if the usual rules of algebra apply to real numbers, then you might notice 10a = 9.999999999999999... and so 10a-9 = .999999999999999... That is, 10a-9 = a. Doing a little algebra shows a=1. That is,

1=.999999999999999...

This is alarming. The average person on the street will swear .999999999999999... is a wee bit smaller than 1. Can they really be equal? While it is true that an infinite sequence of 9's equals 1, the above calculation isn't on solid ground. In one of my first essays for 3QD I criticized a similar shady sleight-of-hand argument which supposedly shows

-1/12 = 1+2+3+4+5+6+7+8+9+10+...

In both cases, the truth is much more interesting. And in both cases, the real question is what do we mean when we write the right-hand side of the equation? In that essay I discussed how we have come to agree on a common understanding of how to interpret infinite sums [4]. This is the beautiful but much-maligned theory of infinite series usually taught in third term Calculus classes.

How about an infinite decimal? How do we interpret such a thing as a number?

My teacher may have explained it, but it was certainly lost on me that writing a number as a decimal is a short-hand for a sum. When we write 2.35817 what we really mean is 2 + 3/10 + 5/100 + 8/1000 + 1/10000 + 7/100000 [5]. If the decimal is infinite, then so is the sum.

Disturbingly, we've just discovered a decimal isn't a number. Rather, it's a shorthand which points to a number. To correctly decode to which number a decimal represents involves computing a potentially infinite sum. We were doing Calculus in fourth grade! It is by doing this infinite sum that we see 0.9999999... and 1.00000... both point to the same number. It seems the real numbers are more subtle than we were led to believe.

There is one bit of good news which comes out of writing real numbers as decimals: we see we can approximate a real number as closely as we like using a rational number. Even better, I can tell you exactly how to find the nearby rational number. All you need to do is take enough digits to be within the allowed error and toss the rest. For example, one of my favorite non-rational numbers is pi:

π=3.14159265358979323846264338327950288419716939937510...

Since π is irrational, its decimal expression goes on forever without repeating itself. Using fast algorithms and fast computers, we have computed more digits then any reasonable person could possibly want. The current record is Peter Trueb's 2016 calculation of 22+ trillion digits of π. But if you are okay with using a decent approximation, then the rational number

3.1415926535897932384626433832795028841971

should suffice. This is already enough to calculate the circumference of the observable universe with an error smaller than an atom!

Since the rational numbers are arbitrarily close to any number on the real number line, it is reasonable to think they pretty much fill the number line. It is awfully tempting to say nearly all real numbers are rational. Actually, quite the opposite is true! 130 years ago Cantor gave a clever argument which showed there are more real numbers between 0 and 1 on the real line then there are rational numbers.

But maybe things aren't so bad. I don't mind non-rational numbers if they come up in a reasonable way. For example, √2 appears as the solution to the equation x²=2. What if we include all real numbers which appear as the solution of a polynomial equation which can be written using rational numbers? This includes all rational numbers, all roots of rational numbers, all numbers which can be constructed using a straightedge and compass (we discussed these numbers here), and much more besides. It doesn't include π or Euler's constant, e, but surely this must be most of the real numbers? Not even close! Once again, the tiniest segment of the real line contains more numbers.

Since this is the computer age, maybe we can expand our mind to include every real number which is "computable". That is, we can write a finite computer program which, in a finite amount of time, can compute as many digits of the number as you like. These are called the computable numbers. Given Mr. Trueb's calculations, it is probably no surprise π is a computable number. These also include all the algebraic numbers of the previous paragraph along with many, many others. In short, the computable numbers include all the real numbers which might show up in nature and definitely include all those we have any hope of computing or understanding. The others are no more knowable than an infinite sequence of coin tosses.

Surely, then, the computable numbers must be nearly all the real numbers? Not by a long shot. The computable numbers are countable - meaning they are closer to being the rational numbers than the real numbers. Even taking all the computable numbers, a sliver of the real line contains unimaginably more!

Rather like the humbling "Pale Blue Dot" photo of the Earth from space, we find the real numbers we can know and imagine are but dust sprinkled among the vastness of the number line.

[1] Although the philosophical, careful, and pedantic will argue this point. After all, it takes Bertrand Russell the better part of 100 pages in the Principles of Mathematics to define the integers.

[2] "I'll gladly pay you Tuesday for a hamburger today." - J. Wellington Wimpy.

[3] Image from here.

[4] Briefly, since we don't know how to do an infinite sum, we instead make use of the fact we know how to compute a sum of finitely many numbers. We can calculate what we get if we add the first two numbers, or add the first three numbers, or the first four numbers, etc. Under favorable circumstances, the list of numbers we generate will get ever closer to a unique real number. We expand our notion of adding numbers by declaring the infinite sum is that number. For example, if we do this for 1/2 + 1/4 + 1/8 + 1/16 + .... we get 1/2, 3/4, 7/8, 15/16, ... for our finite sums. This list of numbers can be shown to be getting arbitrarily close to 1. So we declare 1/2 + 1/4 + 1/8 + 1/16 + .... = 1.

[5] Also, we've all agreed to use base 10. That's why there are 10, 100, 1000, etc. in the bottom of the fractions, and why only the digits 0, ...., 9 are used in the decimal. Three-toed sloth mathematicians would probably use base six. Their fractions would have 6, 36, 96, in the bottom, their decimals would have 0, ..., 5 as their digits, and they would fret over the fact that 1 = .5555555555... Changing the base is akin to changing fonts when you write. Its only effect is aesthetic.

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The vast and mysterious real numbers | 3 Quarks Daily

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by Jonathan Kujawa What is a number? Everyone who takes high school math learns about the real numbers. These are our old friends on the number line. You can hardly do classical algebra or geometry without them. We use the real numbers so often we find them comfortable and familiar.

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Sometime in the late Fifties, Roman Vishniac, a pioneer of photomicrography, picked up a knife, cut a thick horizontal slice of skin from his own thumb and photographed it under a microscope using polarized light. The resulting image succeeded on two levels: the scientific parsing of human skin's rich textural terrain; and the chromatic revelation of natural beauty at a visual scale heretofore inaccessible. By subjecting himself to that brief ouch, he was able to expose the intricacies of the body's largest organ and dramatize a new frontier of optical exploration that would grow exponentially as the technology became more sophisticated. Since then, the dual roles of photomicrography-contributing to scientific investigation, and unveiling eye-popping, artistic devils in the details-has expanded, yielding hidden treasures of a microcosmic universe so populous and dense that the planetary universe of outer space seems paltry by comparison. In some ways this universe of the tiny is more forthcoming than outer space with its endless stretches of nothingness between the orbs.

It is almost impossible to contemplate infinity without feeling infinitesimal. The paradoxical effect of trying to wrap our brains around something as vast as the universe is the realization that we will always be tinier than the tiniest subatomic particle. Compare it to a single cell inside our own bodies trying to fathom the infinity of its host. Infinity makes us giddy.

The reach for infinity usually moves toward outer space and all those unimaginable yonders out there. But what of the other direction, the 'infinitesimals' around and under us, so much smaller than our own miniscule selves? What to make of the recently estimated 23 billion tons of microscopic life at ground level and below our feet, packed together like canned sardines in neighborly proximity to us? This macrograph of a mite on a honey bee isn't a spectacular image, but it does suggest reverse infinity: Is there a mite on the mite? If so, is there a mite on that mite? And so on.

It's not necessary to go deep in this direction to make head-turning discoveries. Technological advances in photomicrography allow us to see far into the invisible world of living organisms and matter, to enter a riotous realm teeming with diversity. As Henry David Thoreau said, ' Heaven is under our feet as well as over our heads.' At this end of the infinity spectrum the sheer density of geological and biological matter within a relatively small space contains multiple worlds for exploration, edification, and aesthetic experience, at all levels of magnification.

If drone photography is about optimal framing of a terrain from above, the showy side of photomicrography could be described as a dazzling fashion shoot. The frame becomes secondary to the object of desire. If the drone photographer is a curator, then the photomicrographer is a stylist, dressing his client in the best possible light for his close-up. The use of color fluorescence, or fluorophores which artificially embellish the wee celebrities, are, at the scientific level, tools of differentiation, a form of color-coding, allowing scientists to separate properties, functions and substances within a given image. The aesthetic bi-products of these interventions span a cultural swath all the way from Las Vegas and pop culture to MOMA-worthy abstract art, including detours to Hollywood's CGI animation studios (with monsters no human imagination could possibly conceive). Nikon recognized the potential appeal of this field early on with an annual competition that goes back more than 40 years, Nikon's Small World.

Of all the optical technologies for microscopic study, none is more spectacular than the confocal laser scanning microscope, the instrument of choice for many of the participants in Nikon's competition. Invented by Marvin Minsky (before he moved on to other pinnacles), confocal technology provides details impossible to achieve with lens-based technology. To understand its capabilities, consider the male diving beetle ( acilius sulcatus), specifically the underside of his front tarsus, exposed most spectacularly in the work of Igor Siwanowicz, a Polish-born neurobiologist whose extracurricular interest in mantises led him first to photomacrography (extreme close-ups of tiny, but visible subjects), and then eventually down into the hidden, subvisible depths of confocal photomicrography where he has accessed the entire color spectrum to achieve startling, exacting images of his subjects.

That male beetle is less than an inch long, which relegates his front feet to the domain of photomicrography at whatever level or technique of magnification. The graphic splendor of Siwanowicz's image, the exotic night-blooming flower we take it to be before reading the fine print, is nearly upstaged when we learn that the suction cups (only one of the intricate functional mechanisms displayed in the image) serve one purpose only, to keep the male from sliding off the back of the female when mating.

Another of Siwanowicz's celebrities, the humped bladderwort ( utricularia gibba) is a carnivorous plant he retrieved from a pond in Virginia not far from where I grew up. Measuring in at a whopping 1-2 millimeters, the plant's trap metamorphizes into a photogenic killing machine of extravagant complexity under the photomicrographer's scrutiny. The desmids it preys upon, seen languishing in its maw, are even tinier-animal-like organisms whose multiple shapes are formal enough to show up on any European coat-of-arms.

With all that color dominating the field, sometimes less becomes more, and more of a surprise, as demonstrated by this enchanting, monochromatic capture of the tracheoles (respiratory tubes) of a European honey bee, by Dr. Lorenzo Domenis. The twisting trunks of trees in a classic Japanese woodblock print could not look more artistic than this:

Art of another, more abstract kind can be seen in the photomicrographs of inanimate materials such as chemical compounds, minerals, acids and biochemical substances. The advantage of working with these materials is that they often come with their own luminescence and colors that emerge under certain lighting conditions. This is where abstract beauties strut their stuff-with colors and color combinations, with luminosity, with dynamic structural complexities and confounding geometrical constructions, and with textures and surfaces both strange and familiar. Who would have thought that a crystal of battery leakage could look like an embroidered tapestry? A kinship with drone photography can be found in the need for optimal framing in these works. Lighting conditions can alter multiple aspects of the substances so that the photomicrographer must also cull or curate the images out of a diverse array of visual effects.

Some exciting works in this category were accomplished far from the high-tech labs of scientific research. Justin Zoll crystallizes amino acids, vitamins, aspirin, and other mundane substances at home and produces his images by attaching the camera to a simple high school microscope he bought on eBay and turning on the lights. Depending on your point of view, his works can seem like decorator samples for Prince of Chintz Mario Buatta, or Georgia O'Keefe on steroids.

Landscapes and cityscapes abound in the crystalline realm of photomicrography, some of them evocative of well-known painters. It's not difficult to spot the work of Lionel Feininger in this micrograph of doxorubin in methanol and dimethylbenzenesulfonic acid by the late Lars Bech, a frequent winner at Nikon's Small World, who managed to channel a variety of painterly styles in his chemical works.

Most of the 'landscapes' in this genre are delightfully eerie, such as the mountainscape with 'snow' at the bottom and bare 'earth' at the top. One work looks like a smoggy city at dawn; another like the Alps on acid. This lookalike effect is an important aspect of photomicrography's appeal, as tiny shocks of recognition jog our memory of similar, unrelated images from our own macro scale of vision.

Unlike drone photography, with photomicrography it's harder to separate the art of a subject from its scientific properties. Many of the biological images function on the level of illustration or graphic design rather than art. And yet, the knowledge that these graphics are generated from non-human sources is enough to elicit an aesthetic reaction even of the lowest order. The aesthetic punch here is something new, incorporating the duality of the image as both mysterious (art) and revelatory (science).

As Siwanowicz sees it: ' Although it is not the primary objective of scientific visual data, surprisingly many research-related images have aesthetic merit; to fully appreciate the beauty of those often abstract and surreal forms one needs to approach them with an open mind.'

The only man-made canvases I can think of that approach this duality are the works of Anselm Kiefer, whose art astounds at the primary level of aesthetic impact, and simultaneously draws us inexorably into the properties of the materials he uses-their implied brutality, their fragility, their complicated metaphorical physicality-the supporting players in the thematic impulses of his work.

So close in, it becomes even more apparent that nature operates on algorithmic principles, its matter programmed for repetitions and patterns that occur according to plan, its architecture addressing specific issues just like ours does, its designs turning function into surprising form and texture. A multifunctional beetle foot demonstrates just how all those different algorithms can peacefully coexist, fitting themselves neatly into a gestalt without crowding, or cancelling, each other out. It makes you wonder: What if nature is merely an AI experiment carried out on a random blue planet like ours, by an alien intelligence? A god-player like ourselves? Is this why things can and do go wrong? Algorithms are only human after all...or natural...or alien.

''Nothing's new under the sun.' This old warhorse should finally be put out to pasture in light of our closer scrutiny of nature's genius. The little shop of wonders we can now access is nothing compared to what remains to be discovered and documented for posterity, requiring many more lifetimes than we probably have left on this earth. Will this be the age of postmodern Romanticism, glorifying nature in a new dimension that slowly opens its cabinet of tiny curiosities for us to see? Is there a new aesthetic afoot which fuses the 'aha' of aesthetic experience with that of scientific wonderment? According to Vishniac, ' Philosophically speaking, there's no difference between art and biology. '

Dante Alighieri goes further: ' Nature is the art of God, ' (or that god-player out there in the ether). Thanks to micrographic technology and its visionary disciples, we're one step closer to seeing what he means.
This is the second of two articles on new ways of seeing. The first can be read here.
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by Brooks Riley Sometime in the late Fifties, Roman Vishniac, a pioneer of photomicrography, picked up a knife, cut a thick horizontal slice of skin from his own thumb and photographed it under a microscope using polarized light. The resulting image succeeded on two levels: the scientific parsing of human skin's rich textural terrain; and...

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Andrew Gelman, Sharad Goel, Daniel E. Ho

A group seeking to ban affirmative action has sued Harvard for discriminating against Asian Americans. The core issues won't be resolved by statistics alone.

Just over forty years ago the Supreme Court struck down race-based quotas in school admissions while also upholding the core tenets of affirmative action. In the landmark 1978 decision, Regents of the University of California v. Bakke, Justice Lewis F. Powell, Jr., singled out Harvard's admissions program as an exemplar for achieving diversity and applauded the university's own description of its policy, according to which the "race of an applicant may tip the balance in his favor just as geographic origin or a life spent on a farm may tip the balance in other candidates' cases." As the Supreme Court would later emphasize, such review considered race merely a "factor of a factor of a factor."

Harvard's policy is now being challenged in federal district court in a suit that could reshape the role of race in college admissions.

In past efforts to dismantle affirmative action-from to, most recently, a case brought by two white women against the University of Texas-the plaintiffs have alleged that race-conscious admissions policies hurt white applicants. But courts have consistently held that race may be employed to achieve the educational gains that stem from a diverse student body when considered "holistically," as one among many factors. The latest legal salvo takes a different, and potentially more potent, tack. The litigants argue that Harvard, in its quest for racial diversity, unjustly penalizes a minority group: Asian Americans.

It is easy to get lost in the nuances of the competing arguments and counterarguments, with expert statistical reports and rebuttals clocking in at over 700 pages. (We know: we read them.) Though they don't always admit it, the experts agree on some broad empirical patterns.

Past efforts to dismantle affirmative action alleged harm to whites. The latest salvo takes a different tack, alleging harm to a minority group: Asian Americans.

The main disagreement is one of interpretation: what factors a university can legitimately consider when making admissions decisions. Backed by conservative activist Edward Blum (who also spearheaded the Texas challenge), the plaintiff group, Students for Fair Admissions (SFFA), advocates for a narrow focus on academic achievement. Harvard, on the other hand, has stressed the importance of non-academic factors, including athletics, character, and family connections. The impasse touches on deep statistical and legal questions of discrimination, merit, and the mission of elite universities.

* * *
Asian Americans and Academics
The Harvard case follows a half century of significant demographic shifts in the United States. Since 1970 Asians have increased from less than 1 percent to nearly 6 percent of the U.S. population. After decades of exclusionary policies, immigration reform in 1965 opened the door for Asian immigrants. Those reforms favored skilled and educated workers, helping to create a large pool of high-achieving Asian American students in subsequent generations.

Set against these demographic changes, Harvard admissions data reveal troubling patterns, SFFA argues. Over the past several years, Asian Americans comprised 27 percent of domestic applicants to Harvard but only 22 percent of domestic applicants accepted for admission; and just under 6 percent of Asian American applicants were admitted, compared to 8 percent of whites. But Asian American applicants had academic credentials and extracurricular track records that were, on average, stronger than those of other racial and ethnic groups, including whites.

SFFA alleges that this disparity indicates a discriminatory double standard, with Asians paying an unfair price for their race-in violation of the Civil Rights Act. Harvard counters that academic achievement is not the sole basis for admission. Indeed, the university receives far more applications from students with stellar high-school transcripts and SAT scores than they could possibly admit. For example, Harvard admits about 2,000 students each year, but more than 8,000 domestic applicants for the class of 2019 had perfect grade point averages (even after Harvard calculated the numbers according to its own index, in order to account for inconsistencies in the way high schools report grades). Rather than mechanically accepting academic superstars, Harvard aims for what it calls "distinguishing excellences," taking an expansive view of talent that goes well beyond a student's grades, test scores, and even extracurriculars.

After accounting for these distinguishing excellences, Harvard argues the apparent penalty against Asian American applicants disappears. SFFA argues that consideration of these factors is simply a veiled effort to limit the number of Asians on campus, a way to sidestep the long-standing prohibition against explicit racial quotas.

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"Distinguishing Excellences"
Much, though not all, of the observed disparity in acceptance rates between Asian American and white applicants stems from Harvard's open preference-which it shares with many elite colleges-for strong athletes and the children of alumni, faculty, and donors, all groups that are disproportionately white in Harvard's pool of applicants. Whereas Harvard's overall acceptance rate is about 7 percent, the college admits 86 percent of recruited athletes and 34 percent of "legacy" candidates-those with a parent who attended Harvard. Legacies account for 22 percent of white admits but just 7 percent of Asian American admits.

Yet setting aside legacies and athletes, Asian American applicants are still admitted at lower rates than whites with comparable academic and extracurricular records.

This remaining disparity largely boils down to admissions decisions based on personality, geography, and family. Harvard assesses the "personal" qualities of applicants on a scale from 1 to 6 ("outstanding" to "worrisome"), and on this dimension Asian Americans are rated lower on average than whites. Harvard officials describe "personal quality" as "a subjective determination of a combination of many, many factors," including "perhaps likability, [and] character traits, such as integrity, helpfulness, courage, kindness." The university likewise considers where applicants live and their parents' occupations. Harvard may, for example, favor students from rural communities and disfavor the children of engineers.

* * *
Adjusting and Over-Adjusting for Differences
In many studies of discrimination, race-based or otherwise, an apparent disparity disappears once one accounts for other factors. In Harvard's case, the gap in admissions rates between Asian Americans and whites largely vanishes after adjusting for differences in legacy status, athleticism, personal ratings, geography, and parental occupation.

Even if a variable helps to explain away a disparity between groups, that variable may itself be the product of discrimination.

In assessing whether Harvard intentionally discriminates against Asian applicants, a key question, then, is whether the factors the university uses to guide admissions decisions are themselves appropriate. If personal ratings were awarded in racially discriminatory ways, it would be inappropriate to appeal to them to explain disparities in admissions. Likewise, if personal ratings bear little relation to legitimate educational goals, then differences in admissions rate should not be justified by differences in the ratings.

This statistical issue-where controlling for illegitimate factors masks evidence of discrimination-is an instance of what is sometimes called "included-variable bias" (as opposed to the inverse problem of "omitted-variable bias," which entails leaving out variables that ought to be included). In our own research on stop-and-frisk policing, we find that one can underestimate racial bias by improperly controlling for factors such as an officer's judgment about whether a suspect is behaving "furtively," since such judgments are often related more to race than risk.

In short, models can be misleading not only for the variables they omit, but also for the variables they include. Even if a variable helps to explain away a disparity between groups, that variable may itself be the product of discrimination or have little rational relation to a legitimate policy goal.

* * *
The Evolving Meaning of Merit
To gauge the appropriateness of Harvard's admissions criteria, SFFA emphasizes the historical context of the university's current admissions practices, which sociologist Jerome Karabel has exhaustively traced back to anti-Semitism in the early twentieth century.

At that time, Harvard formally and primarily relied on entrance exams to select students. But alarm arose when Jewish applicants began to be accepted in large numbers, often out-performing their wealthy Protestant counterparts on these exams. Harvard's then-president, A. Lawrence Lowell, controversially proposed capping Jewish enrollment. Instead the university shifted its admissions to focus on the "character" of applicants, as well as geographic diversity and legacy admissions, achieving Lowell's end by using different, less explicit means.

Harvard argues its current admissions practices are distinct both in form and in function from those of its discriminatory past. Karabel himself disclaims the relevance of former anti-Semitism to the SFFA lawsuit. But this history can unsettle ideals of well-roundedness and the need for holistic admissions to make suitably individualized decisions.

When it comes to discrimination, the relationship between intentions and effects is a subtle but important one. As Harvard has previously indicated, its contemporary preference for legacy students is intended to foster the alumni community and encourage donations, not to curb admission of racial and ethnic minorities. Nevertheless, favoring legacies benefits a disproportionately large subset of white applicants while leaving less space for the vast majority of black, Hispanic, and Asian applicants without family ties to the university, raising concerns about equity.

A similar problem plagues the university's focus on athletics. Whereas holistic admissions originated in anti-Semitism, values of student athleticism are rooted in an ideology of "muscular Christianity," the notion that manliness and moral leadership go hand in hand. When former Harvard president Charles Eliot threatened in 1905 to abolish football for its brutality-player deaths were not uncommon at the time-Theodore Roosevelt intervened, pledging to make the sport safer without it becoming "too ladylike." In contrast to university systems abroad, American colleges place a marked emphasis on athleticism and the character-building aspects of organized sports.

Broadcast collegiate sports may be racially diverse, but athletic preferences at elite colleges appear mostly to benefit white applicants. At Harvard, about three-quarters of athletes admitted are white. As Saahil Desai has written in The Atlantic, selecting students for lacrosse, golf, sailing, and water polo may be a "quiet sort of affirmative action for affluent white kids."

Despite these critiques, the U.S. Department of Education's Office of Civil Rights has previously concluded that it is permissible for colleges to give preferential treatment to both athletes and legacies. It is unlikely that view will change anytime soon. (President Trump's Department of Justice did file a brief in the Harvard case, but while mentioning athletic and legacy preferences, it took no position on them.) SFFA largely concedes this point, focusing its argument on the disparity that remains after adjusting for differences in athletic recruitment and family connections.

* * *
Character and Bias
The use of personal ratings is one of the most controversial aspects of Harvard's admissions process. Regardless of whether one believes Harvard should consider personal traits, the measure itself may have problems, stemming in part from an inevitable entangling of subjective judgment, bias, and larger goals in the admissions process.

If personal ratings were awarded in racially discriminatory ways, it would be inappropriate to appeal to them to explain disparities in admissions.

The personal ratings look different depending on who awards them: alumni interviewers or Harvard's internal admissions staff. On average, alumni give white and Asian American applicants similar ratings, but Harvard staff give whites substantially better reviews than they give Asian Americans. For the alumni-assigned ratings, 50 percent of Asian American applicants and 51 percent of whites were rated as having "very strong" or "outstanding" personal traits. But for personal ratings awarded by Harvard's internal admissions staff, only 18 percent of Asian Americans were in the top group, compared to 23 percent of whites. White applicants received these top ratings about 30 percent more often than Asian Americans.

With acceptance rates well under 10 percent, most students have little chance of admission to Harvard. So it is important to look as well at the subset of academically competitive students, for whom personal scores may matter more. The graphs below show the percentage of applicants with strong personal ratings-in the top two categories of "very strong" or "outstanding"-broken down by academic performance.

On average, students with higher academic ratings get higher personal ratings. This pattern holds for all racial and ethnic groups and for both staff and alumni evaluations-and is perhaps worth studying in its own right.

Among the most competitive applicants, the graphs show that Harvard staff are still more likely than alumni to rate whites more favorably than Asian Americans. In the top academic decile of applicants, Harvard staff put 23 percent of Asian Americans and 31 percent of whites in the top two personality categories. In contrast, alumni interviewers gave this group of Asian and white applicants these top ratings at much closer rates (64 percent and 66 percent, respectively).

SFFA argues that these patterns indicate that Harvard staff manipulated the ratings or were otherwise biased against Asian American applicants. After all, subjective assessments such as "leadership potential," "courage," "grit," and "fit" are particularly prone to subtle forms of discrimination. Harvard, on the other hand, claims its staff reviewers are simply more qualified than alumni interviewers to make such judgments, in part because they have access to more detailed information than the alumni.

* * *
A Path Forward
Adjudicating the competing empirical claims based on the expert reports alone is difficult for academic researchers, let alone judges. But we see three issues worth highlighting.

The Empirical Must Not Obscure the Normative
First, empirical questions must not obscure the more fundamental normative issues. Whether experts should adjust for factors such as personal ratings in their statistical models depends on normative assessments of how these criteria relate to Harvard's educational mission. Consider that one of the contested factors is whether a domestic applicant was born in the United States. One might favor foreign-born applicants as a way of making the student body more geographically diverse, or disfavor such individuals to create a college cohort that better resembles a cross-section of Americans. Yet foreign-born status can also be a proxy for race. Qualitative grounding of statistical findings is important to avoid misleading conclusions; it is not enough to control for a factor in a statistical analysis without clarifying its role in admissions goals and educational objectives.

Warring Expert Reports Do Not Serve Justice
Second, when legal judgments rest on complex empirical analysis, courts are ill-served by warring expert reports. The two sides in this case analyzed different subsets of the data and used somewhat different statistical methods, making direct comparisons difficult. For example, some of the plaintiff's analyses excluded applicants who applied early, and Harvard analyzed each yearly cohort of applicants in isolation. Both reports perform some analyses on all categories of applicants and some analyses that exclude special categories (e.g., legacies and athletes) that constitute a small fraction of applicants but a large proportion of admitted students. Such slicing and dicing can be problematic.

Our preference would have been to analyze all (domestic) applicants together, but in a way that would allow one to detect a potential pattern of discrimination that may differ across subpopulations-including groups defined by application cycle, ancestry, grades, and test scores. We have used that sort of analysis, termed multilevel modeling, many times in our own work, including to understand how the relation between income and voting varies by geography.

As we have learned from the replication crisis sweeping the biomedical and social sciences, it is frighteningly easy for motivated researchers working in isolation to arrive at favored conclusions-whether inadvertently or intentionally. As a partial solution, former federal judge Richard Posner has suggested that courts appoint independent experts to sift through the statistical evidence.

Admissions Criteria Should Be Made Clearer
Third, Harvard and other universities could work to better codify and align admissions procedures with institutional goals. If parental occupation, for instance, is merely a proxy for socioeconomic status, more direct measurement of socioeconomic status may reduce unwarranted discretion. The risk of such clearer, more parsimonious admissions criteria is that they may expose the value placed on race, running up against the Supreme Court's distaste for race-conscious point systems in college admissions.

SFFA argues that Harvard should simply replace its affirmative action policy with an admissions plan that favors students from disadvantaged socioeconomic backgrounds. In response, Harvard states that such alternative approaches "either are ineffective at generating a racially diverse class, or would significantly alter the composition of the admitted class along other dimensions." Regardless, greater clarity in admissions criteria would help resolve statistical disagreements.

* * *
The Future of Affirmative Action
The core contention in this case is about the meaning of merit and its role in elite education. Both sides agree that even if one sets aside legacy applicants and recruited athletes, Asian Americans are admitted at lower rates than whites with similar academic and extracurricular credentials. Less clear is the role that "multi-dimensional excellence" (read: athleticism and "outstanding" personality), family background, and geographic diversity should play in college admissions-and accordingly in statistical assessments of bias-particularly if a focus on these factors favors whites over Asians.

The substantive legal and policy questions here have more to do with potential anti-Asian bias than with affirmative action itself.

SFFA's stated goal is to end affirmative action. Harvard has defended it, triggering progressive support for race-conscious admissions and, at the same time, for Harvard's side in this lawsuit. Yet those two things-affirmative action, on the one hand, and Harvard's particular admissions practices, on the other-are not one and the same. The substantive legal and policy questions here have more to do with potential anti-Asian bias than with affirmative action itself. As a dean at the University of California, Berkeley, has said, Asian Americans are "being used as a pawn in a chess game." Even if SFFA is right that Harvard's admissions practices are biased against Asians, that does not mean affirmative action itself is to blame.

As a result, if the courts do find that Harvard has improperly discriminated against Asian applicants, one remedy is to simply curb that practice; they need not curtail long-standing affirmative action policies that increase the representation of underrepresented minorities on college campuses.
Editors' Note: Gelman and Ho received their PhDs from Harvard, and Gelman has taught at Harvard as well. None of the authors has been involved in the SFFA lawsuit against Harvard, and they have no financial conflicts of interests.

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what statistics can't tell us in the fight over affirmative action at harvard

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What Statistics Can't Tell Us in the Fight over Affirmative Action at Harvard

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What Statistics Can't Tell Us in the Fight over Affirmative Action at Harvard from Boston Review. A group seeking to ban affirmative action has sued Harvard for discriminating against Asian Americans. The core issues won't be resolved by statistics alone.

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A truly amazing and beautiful connection between math and physics | 3 Quarks Daily

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Please watch videos in order. A good understanding of high school math and physics is required. [Thanks to Jessica Collins for showing me this.]

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https://aeon.co/ideas/the-concept-of-probability-is-not-as-simple-as-you-think

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The concept of probability is not as simple as you think | Aeon Ideas

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