"How Not to Be Wrong"

[fiefoe]

Another MY KID MUST READ THIS LATER book. Jordan Ellenberg is a remarkably lucid and deft writer. Just the section on Buffon’s needle problem was worth the price of admission; understanding the proof gave me such a jolt of pleasure at that a-ha moment.

• One thing the American defense establishment has traditionally understood very well is that countries don’t win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost. That’s not the stuff war movies are made of, but it’s the stuff wars are made of.
• Why did Wald see what the officers, who had vastly more knowledge and understanding of aerial combat, couldn’t? It comes back to his math-trained habits of thought. A mathematician is always asking, “What assumptions are you making? And are they justified?”
• this is just the introduction. When you write a book explaining human reproduction to preteens, the introduction stops short of the really hydraulic stuff about how babies get inside Mommy’s tummy.
• If there’s an optimal answer, it’s somewhere in the middle, and deviating from it in either direction is bad news. Nonlinear thinking means which way you should go depends on where you already are.
• The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we’d now call mathematics, things we’d now call religion, and things we’d now call mental illness. They believed that odd numbers were good and even numbers evil; that a planet identical to our own, the Antichthon, lay on the other side of the sun; and that it was wrong to eat beans, by some accounts because they were the repository of dead people’s souls.
• This is not just loosey-goosey mathematical relativism. Just because we can assign whatever meaning we like to a string of mathematical symbols doesn’t mean we should. In math, as in life, there are good choices and there are bad ones. In the mathematical context, the good choices are the ones that settle unnecessary perplexities without creating new ones.
• We can’t hold on to both of these desires at once; one must be discarded. In Cauchy’s approach, which has amply proved its worth in the two centuries since he invented it, it’s the uniqueness of the decimal expansion that goes out the window.
• See the problem? If all Americans are supposed to be overweight in 2048, where are those one in five future black men without a weight problem supposed to be? Offshore? That basic contradiction goes unmentioned in the paper. It’s the epidemiological equivalent of saying there are −4 grams of water left in the bucket. Zero credit.
• That’s how the Law of Large Numbers works: not by balancing out what’s already happened, but by diluting what’s already happened with new data, until the past is so proportionally negligible that it can safely be forgotten.
• The slogan to live by here is: Don’t talk about percentages of numbers when the numbers might be negative.
• It’s only after you’ve started to formulate these questions that you take out the calculator. But at that point the real mental work is already finished. Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.
• Never underestimate the territorial ambitions of mathematics! You want to know about God? There are mathematicians on the case.
• modern people with a scientific worldview were presented with an unexpected avenue toward accepting religious faith, and many took it. I have it on good assurance that one new father from a secular Jewish family waited until the Statistical Science paper was officially accepted before deciding to circumcise his son.
• But if you tweak the game, making it less clearly fraudulent but leaving unchanged the potential to mislead, you find the Baltimore stockbroker is alive and well in the financial industry... The life of an incubated fund is not as warm and safe as the name might suggest. Typically, companies incubate lots of funds at once, experimenting with numerous investment strategies and allocations. The funds jostle and compete in the womb. Some show handsome returns, and are quickly made available to the public, with extensive documentation of their earnings so far. But the runts of the litter are mercy-killed, often without any public notice that they ever existed... But the data suggests the opposite: the incubator funds, once the public gets their hands on them, don’t maintain their excellent prenatal performance, instead offering roughly the same returns as the median fund... When you sink your savings into the incubated fund with the eye-popping returns, you’re like the newsletter getter who invests his life savings with the Baltimore stockbroker; you’ve been swayed by the impressive results, but you don’t know how many chances the broker had to get those results.
• The Baltimore stockbroker con works because, like all good magic tricks, it doesn’t try to fool you outright. That is, it doesn’t try to tell you something false—rather, it tells you something true from which you’re likely to draw incorrect conclusions.
• The mistake is in being surprised by this encounter with the improbable. The universe is big, and if you’re sufficiently attuned to amazingly improbable occurrences, you’ll find them. Improbable things happen a lot.
• Rather, McKay and Bar-Natan are making a potent point about the power of wiggle room. Wiggle room is what the Baltimore stockbroker has when he gives himself plenty of chances to win; wiggle room is what the mutual fund company has when it decides which of its secretly incubating funds are winners and which are trash.
• Suppose the null hypothesis is true, and let p be the probability (under that hypothesis) of getting results as extreme as those observed. The number p is called the p-value. If it is very small, rejoice; you get to say your results are statistically significant. If it is large, concede that the null
• And yet we think we know what to expect, thanks to a remarkably fruitful point of view: we think of primes as random numbers. The reason the fruitfulness of this viewpoint is so remarkable is that the viewpoint is so very, very false.
• And a lot of twin primes are exactly what number theorists expect to find, no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt.
• The Goldbach conjecture, that every even number greater than 2 is the sum of two primes, is another one that would have to be true if primes behaved like random numbers. So is the conjecture that the primes contain arithmetic progressions of any desired length, whose resolution by Ben Green and Terry Tao in 2004 helped win Tao a Fields Medal.
• Building on the work of many predecessors, Zhang is able to prove that the prime numbers look random in the first way we mentioned, concerning the remainders obtained after division by many different integers. From there,* he can show that the prime numbers look random in a totally different sense, having to do with the sizes of the gaps between them. Random is random!
• But it’s all worth it for those moments of discovery, where everything works, and you find that the texture and protrusions of the liver really do predict the severity of the following year’s flu season, and, with a silent thank-you to the gods, you publish. You might find this happens about one time in twenty. That’s what I’d expect, anyway. Because I, unlike you, don’t believe in haruspicy.
• In other words, among the “OMG I found the schizophrenia gene” results that might get published, there are five hundred times as many bogus ones as real ones. And that’s assuming that all the genes that really do have an effect on schizophrenia pass the test! As we saw with Shakespeare and basketball, it’s very possible for a real effect to be rejected as statistically insignificant if the study isn’t high powered enough to find it.
• that means that of the genes certified by p-value to cause schizophrenia, only five really do so, as against the five thousand pretenders that passed the test by luck alone.
• You can see from the picture that the significance test isn’t the problem. It’s doing exactly the job it’s built to do. The genes that don’t affect schizophrenia very rarely pass the test, while the genes we’re really interested in pass half the time. But the nonactive genes are so massively preponderant that the circle of false positives, while small relative to the true negatives, is much larger than the circle of true positives.
• But the relatively small size of the study means a more realistic assessment of the strength of the effect would have been rejected, paradoxically, by the p-value filter. In other words, we can be quite confident that the large effect reported in the study is mostly or entirely just noise in the signal. But noise is just as likely to push you in the opposite direction from the real effect as it is to tell the truth. So we’re left in the dark by a result that offers plenty of statistical significance but very little confidence. Scientists call this problem “the winner’s curse,”
• But everybody knows it’s not really right. When they don’t think anyone’s listening, scientists call this practice “torturing the data until it confesses.” And the reliability of the results are about what you’d expect from confessions extracted by force.
• The theory U acts as a kind of Bayesian coating to T, keeping new evidence from getting to it and dissolving it. This is a property most successful crackpot theories have in common; they’re encased in just enough protective stuff that they’re equally consistent with many possible observations, making them hard to dislodge. They’re like the multi-drug-resistant E. coli of the information ecosystem.
• Given that we actually do exist, so that the truth is in the bottom row, almost all the probability is sitting in SIMS. Yes, the existence of human life is evidence for the existence of God; but it’s much better evidence that our world was programmed by people much smarter than us.
• It starts with the game of franc-carreau, which, like the Genoese lottery, reminds you that people in olden times would gamble on just about anything. All you need for franc-carreau is a coin and a floor with square tiles. You throw the coin on the floor and make a bet: will it land wholly within one tile, or end up touching one of the cracks?
• Buffon solved this equation (and so can you, if that’s the kind of thing you’re into), finding that franc-carreau was a fair game just when the side of the carreau was 4 + 2√2 times the radius of the coin, a ratio of just under 7. This was conceptually interesting, in that the combination of probabilistic reasoning with geometric figures was novel;
• the needle crosses a crack substantially more often than it lands wholly within a single slat. Buffon’s needle problem has a beautifully unexpected answer: the probability is 2 / π, or about 64%... Let’s start small. What if we ask, more generally, about the expected number of crack crossings by a needle that’s two slats wide?... In other words, the formula Expected number of crossings of a length-N needle= Np holds for bent needles, too. Here’s one such needle:... Which means the expected number of crossings of the diameter-1 circle is just about exactly the same as the expected number of crossings of the 65,536-gon. And by our bent-needle rule, that’s Np, where N is the perimeter of the polygon. What’s that perimeter? It must be almost exactly that of the circle; the circle has radius 1/2, so its circumference is π. So the expected number of times the circle crosses a crack is πp.
• Because how many crossings does the circle have? All of a sudden, what looked like a hard problem becomes easy. The symmetry we lost when we went from coin to needle has now been restored by bending the needle into a circular hoop. And this simplifies matters tremendously. It doesn’t matter where the circle falls—it crosses the lines in the floor exactly twice... So the expected number of crossings is 2; and it is also πp; and so we have discovered that p= 2 / π, just as Buffon said. In fact, the argument above applies to any needle, however polygonal and curvy it might be; the expected number of crossings is Lp, where L is the length of the needle in slat-width units. Throw a mass of spaghetti on the tile floor and I can tell you exactly how many times to expect a strand to cross a line. This generalized version of the problem is called, by mathematical wags, Buffon’s noodle problem.
• Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.
• With the unexpected infusion of cash from Random Strategies, the jackpot on Monday, August 16, stood at $2.1 million. It was a roll-down, payday for lottery players, and nobody except the MIT students knew it was coming. Almost 90% of the tickets for the drawing were held by Harvey’s team. They were standing in front of the money spigot, all alone. And when the drawing was over, Random Strategies had made $700,000 on their $1.4 million investment, a cool 50% profit.
• Eventually, the state caught on and canceled the program, but not before La Condamine and Voltaire had taken the government for enough money to be rich men for the rest of their lives. What—you thought Voltaire made a living writing perfectly realized essays and sketches?
• “What’s the right amount of the taxpayers’ money to be wasting?” To paraphrase Stigler: if your government isn’t wasteful, you’re spending too much time fighting government waste.
• The two men stood at odds across a gulf as much temperamental as philosophical. Voltaire’s generally chipper outlook had no room for Pascal’s dark, introspective, mystical emissions. Voltaire dubbed Pascal “the sublime misanthrope” and devoted a long essay to knocking down the gloomy Pensées piece by piece. His attitude toward Pascal is that of the popular smart kid toward the bitter and nonconforming nerd.
• The younger Bernoulli’s beautiful untwisting of the paradox is a landmark result, and one that has formed the foundation of economic thinking about uncertain values ever since. The mistake, Bernoulli said, is to say that a ducat is a ducat is a ducat. A ducat in the hand of a rich man is not worth the same as a ducat in the hand of a peasant, as is plainly visible from the different levels of care with which the two men treat their cash.
• In fact, he’d been picking at the foundations of game theory since his undergraduate senior thesis. At RAND, he devised a famous experiment now known as Ellsberg’s paradox... BLACK, on the other hand, subjects the player to an “unknown unknown”—not only are we not sure whether the ball will be black, we don’t have any knowledge of how likely it is to be black. In the decision-theory literature, the former kind of unknown is called risk, the latter uncertainty. Risky strategies can be analyzed numerically; uncertain strategies, Ellsberg suggested, were beyond the bounds of formal mathematical analysis, or at least beyond the bounds of the flavor of mathematical analysis beloved at RAND.
• And it’s not coincidence that projective geometry arose naturally from attempts to solve the practical problem of depicting the three-dimensional world on a flat canvas. Mathematical elegance and practical utility are close companions, as the history of science has shown again and again. Sometimes scientists discover the theory and leave it to mathematicians to figure out why it’s elegant, and other times mathematicians develop an elegant theory and leave it to scientists to figure out what it’s good for. One thing the projective plane is good for is representational painting. Another is picking lottery numbers.
• For example: here’s Fano’s plane again, but with the points labeled by the numbers 1 through 7: Look familiar? If we list the seven lines, recording for each the set of three points that constitute it, we get: 124 135 167 257 347 236 456 ... This is none other than the seven-ticket combo we saw in the last section, the one that hits each pair of numbers exactly once, guaranteeing a minimum payoff. At the time, that property seemed impressive and mystical. How could anyone have come up with such a perfectly arranged set of tickets? But now I’ve opened the box and revealed the trick: it’s simple geometry. Each pair of numbers appears on exactly one ticket, because each pair of points appears on exactly one line. It’s just Euclid, even though we’re speaking now of points and lines Euclid would not have recognized as such.
• Lojban, one of the most successful contemporary examples,* has a strict rule that no two of the basic roots, or gismu, are allowed to be too phonetically close.
• Hamming’s notion of “distance” follows Fano’s philosophy—a quantity that quacks like distance has the right to be called distance. But why stop there? The set of points at distance less than or equal to 1 from a given central point has a name in Euclidean geometry; it is called a circle, or, if we are in higher dimensions, a sphere.* So we’re compelled to call the set of strings at Hamming distance at most 1* from a code word a “Hamming sphere,” with the code word at the center. For a code to be an error-correcting code, no string—no point, if we’re to take this geometric analogy seriously—can be within distance 1 of two different code words; in other words, we ask that no two of the Hamming spheres centered at the code words share any points. So the problem of constructing error-correcting codes has the same structure as a classical geometric problem, that of sphere packing: how do we fit a bunch of equal-sized spheres as tightly as possible into a small space, in such a way that no two spheres overlap? More succinctly, how many oranges can you stuff into a box?
• Does the geometric story of high-dimensional sphere packings give insight into the theory of error-correcting codes, as the geometric story of the projective plane did? In this case, the flow has mostly been in the other direction;* the insights from coding theory have instigated progress in sphere packings.
• but here’s the thing: any mathematical object as startling as the Leech lattice is bound to be important. It turned out that the Leech lattice was very rich in symmetries of a truly exotic kind. The master group theorist John Conway, upon encountering the lattice in 1968, worked out all its symmetries in a twelve-hour spree of computation on a single giant roll of paper. These symmetries ended up forming some of the final pieces of the general theory of finite symmetry groups that preoccupied algebraists for much of the twentieth century.*
• When you encounter an intricate construction like Hamming’s, you’re naturally inclined to think an error-correcting code is a very special thing, designed and engineered and tweaked and retweaked until every pair of code words has been gingerly nudged apart without any other pair being forced together. Shannon’s genius was to see that this vision was totally wrong. Error-correcting codes are the opposite of special. What Shannon proved—and once he understood what to prove, it was really not so hard—was that almost all sets of code words exhibited the error-correcting property; in other words, a completely random code, with no design at all, was extremely likely to be an error-correcting code. This was a startling development, to say the least. Imagine you were tasked with building a hovercraft; would your first approach be to throw a bunch of engine parts and rubber tubing on the ground at random, figuring the result would probably float?
• And in the decades between then and now, mathematicians have tried to ride that conceptual boundary between structure and randomness, laboring to construct codes random enough to be fast, but structured enough to be decodable.
• The guaranteed returns of the Denniston strategy are replaced by risk. Naturally, that risk comes with an upside, too—team Selbee has a 32% chance of getting more than six of those prizes, impossible if you pick your tickets according to Denniston. The expected value of Selbee’s tickets is the same as that of Denniston’s, or anyone else’s. But the Denniston method shields the player from the winds of chance. In order to play the lottery without risk, it’s not enough to play hundreds of thousands of tickets; you have to play the right hundreds of thousands of tickets.
• The graph reflects a sobering social fact, which is by now commonplace in the political science literature. Undecided voters, by and large, aren’t undecided because they’re carefully weighing the merits of each candidate, unprejudiced by political dogma. They’re undecided because they’re barely paying attention.
• Keep this in mind when you’re told that two phenomena in nature or society were found to be uncorrelated. It doesn’t mean there’s no relationship, only that there’s no relationship of the sort that correlation is designed to detect.
• To put it in words: if you’re in the hospital, you’re there for a reason. If you’re not diabetic, that makes it more likely the reason is high blood pressure. So what looks at first like a causal relationship between high blood pressure and diabetes is really just a statistical phantom.
• You might also guess that, since the slime mold found 3-dark and 5-light equally attractive before, it would continue to do so in the new context. In the economist’s terms, the presence of the new option shouldn’t change the fact that 3-dark and 5-light have equal utility. But no: when 1-dark is available, the slime mold actually changes its preferences, choosing 3-dark more than three times as often as it does 5-light!
• The mathematical buzzword in play here is “independence of irrelevant alternatives.” That’s a rule that says, whether you’re a slime mold, a human being, or a democratic nation, if you have a choice between two options, A and B, the presence of a third option, C, shouldn’t affect which of A and B you like better.
• On the other hand, he became quickly exasperated with people whose intellectual standards didn’t match his own. This combination of timidity and temper led his mentor Jacques Turgot to nickname him “le mouton enragé,” or “the rabid sheep.”
• In 1820, the Hungarian noble Farkas Bolyai, who had given years of his life to the problem without success, warned his son János against following the same path: You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone. . . . I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. . . . I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example. . . .
• In this view, the hard questions about law, the ones that make it all the way to the Supremes, are left indeterminate by the axioms. The justices are thus in the same position Pascal was when he found he couldn’t reason his way to any conclusion about God’s existence. And yet, as Pascal wrote, we don’t have the choice not to play the game. The court must decide, whether it can do so by conventional legal reasoning or not. Sometimes it takes Pascal’s route: if reason does not determine the judgment, make the judgment that seems to have the best consequences.
• As the philosopher W. V. O. Quine put it, “To believe something is to believe that it is true; therefore a reasonable person believes each of his beliefs to be true; yet experience has taught him to expect that some of his beliefs, he knows not which, will turn out to be false. A reasonable person believes, in short, that each of his beliefs is true and that some of them are false.” Formally, this is very similar to the apparent contradictions in American public opinion we unraveled in chapter 17. The American people think each government program is worthy of continued funding, but that doesn’t mean they think all government programs are worthy of continued funding.
• We are not free to say whatever we like about the wild entities we make up. They require definition, and having been defined, they are no more psychedelic than trees and fish; they are what they are. To do mathematics is to be, at once, touched by fire and bound by reason. This is no contradiction. Logic forms a narrow channel through which intuition flows with vastly augmented force.
• It’s a little reminiscent of Orson Scott Card’s short story “Unaccompanied Sonata,” which is about a musical prodigy who is carefully kept alone and ignorant of all other music in the world so his originality won’t be compromised, but then a guy sneaks in and plays him some Bach, and of course the music police can tell what happened, and the prodigy ends up getting banished from music, and later I think his hands get cut off and he’s blinded or something, because Orson Scott Card has this weird ingrown thing about punishment and mortification of the flesh, but anyway, the point is, don’t try to keep young musicians from hearing Bach, because Bach is great.