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June 25 - July 20, 2019
The military analyst has two options for explaining this: either the German bullets just happen to hit every part of the plane but one, or the engine is a point of total vulnerability. Both stories explain the data, but the latter makes a lot more sense. The armor goes where the bullet holes aren’t.
A mathematician is always asking, “What assumptions are you making? And are they justified?”
To a mathematician, the structure underlying the bullet hole problem is a phenomenon called survivorship bias
squaring! A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.
What’s the numerical value of an infinite sum? It doesn’t have one—until we give it one. That was the great innovation of Augustin-Louis Cauchy, who introduced the notion of limit into calculus in the 1820s.*
Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.
An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something’s wrong with your method.
Law of Large Numbers.
the more coins you flip, the more and more extravagantly unlikely it is that you’ll get 80% heads. In fact, if you flip enough coins, there’s only the barest chance of getting as many as 51%!
The smaller the number of coins—what we’d call in statistics the sample size—the greater the variation in the proportion of heads.
De Moivre’s insight is that the size of the typical discrepancy* is governed by the square root of the number of coins you toss.
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.
Don’t talk about percentages of numbers when the numbers might be negative.
It’s easy to think of the quantitative analysis of policy as something you do with a calculator. But the calculator enters only once you’ve figured out what calculation you want to do.
Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.
The universe is big, and if you’re sufficiently attuned to amazingly improbable occurrences, you’ll find them. Improbable things happen a lot.
In the British statistician R. A. Fisher’s famous formulation, “the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us.”
Inference is a hard thing, maybe the hardest thing. From the shape of the clouds and the way they move we struggle to go backward, to solve for x, the system that made them.
The world is so richly structured and so perfectly ordered—how tremendously unlikely it would be for there to be a world like this one, under the null hypothesis that there’s no primal designer who put the thing together!
The test of significance does not provide the information concerning psychological phenomena characteristically attributed to it . . . a great deal of mischief has been associated with its use. . . . To say it “out loud” is, as it were, to assume the role of the child who pointed out that the emperor was really outfitted only in his underwear.
And less-effective birth control means more pregnancies. (What—you thought I was going to say there was a wave of abstinence?)
A high-powered study, like the birth control trial, may lead you to burst a vein about a small effect that isn’t actually important. An underpowered one may lead you to wrongly dismiss a small effect that your method was simply too weak to see.
Genomicists nowadays believe that heritability of IQ scores is probably not concentrated in a few smarty-pants genes, but rather accumulates from numerous genetic features, each one having a tiny effect.
dopes. In part, the replicability crisis is simply a reflection of the fact that science is hard and that most ideas we have are wrong—even most of those ideas that survive a first round of prodding.
The confidence interval is the range of hypotheses that the reductio doesn’t demand that you trash, the ones that are reasonably consistent with the outcome you actually observed.
A statistically significant finding gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge.
Just as the prior describes your beliefs before you see the evidence, the posterior describes your beliefs afterward. What we’re doing here is called Bayesian inference, because the passage from prior to posterior rests on an old formula in probability called Bayes’s Theorem.
In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.
When Fisher says that “no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas,” he is saying exactly that scientific inference can’t, or at least shouldn’t, be carried out purely mechanically; our preexisting ideas and beliefs must always be allowed to play a part.
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.
“Some have argued that it’s highly unlikely for something as complex as the Earth’s biosphere to have arisen purely by natural selection without any intervention from outside. By far the most likely such explanation is that we are actually not physical beings at all, but residents of a computer simulation being carried out by humans with unthinkably advanced technology, to what purpose we can’t exactly know. It’s also possible that we were created by a community of gods, something like those worshiped by the ancient Greeks. There are even some people who believe that one single God created the
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Rather, I take the queasy feeling these arguments generate as an indication that we’ve reached the limits of quantitative reasoning.
Powerball isn’t the only lottery, but all lotteries have one thing in common; they’re bad bets. A lottery, just as Adam Smith observed, is designed to return a certain proportion of ticket sales to the state; for that to work, the state has to take in more money in tickets than it gives out in prizes.
“The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration . . . the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it . . . yet it finally surrounds the resistant substance.”
Math, like meditation, puts you in direct contact with the universe, which is bigger than you, was here before you, and will be here after you. It might drive me crazy not to do it.
Everlasting joy in return for one day’s effort on earth.
The doctrine of expected utility is appealingly straightforward and simple: presented with a set of choices, pick the one with the highest expected utility.
The rich person, who has enough reserves to absorb those occasional losses, invests and gets richer; the nonrich people stay right where they are.
Understand this: I warmly endorse, in fact highly recommend, a bristly skepticism in the face of all claims that such-and-such an entity can be explained, or tamed, or fully understood, by mathematical means.
And yet the history of mathematics is a history of aggressive territorial expansion, as mathematical techniques get broader and richer, and mathematicians find ways to address questions previously thought of as outside their domain.
In the Kahneman-Tversky theory, people tend to place more weight on low-probability events than a person obedient to the von Neumann-Morgenstern axioms would; so the allure of the jackpot exceeds what a strict expected utility calculation would license.
By means of the eccentricity—a geometric quantity at least as old as the work of Apollonius of Perga in the third century BCE—Galton had found a way to measure the association between two variables, and in so doing had solved a problem at the cutting edge of nineteenth-century biology: the quantification of heredity.
The presence of correlation makes compression possible; actually doing it involves much more modern ideas,
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.
Clinical researchers call this the surrogate endpoint problem. It’s time consuming and expensive to check whether a drug improves average life span, because in order to record someone’s life span you have to wait for them to die. HDL level is the surrogate endpoint, the easy-to-check biomarker that’s supposed to stand in for “long life with no heart attack.”
And if we held ourselves to a stricter evidentiary standard, declining to issue any of these recommendations because we weren’t sure we were right? Then the lives we would have saved would be lost instead.
But one thing’s for certain: refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy. It’s a lot like George Stigler’s advice about missing planes. If you never give advice until you’re sure it’s right, you’re not giving enough advice.
Public opinion doesn’t exist. More precisely, it exists sometimes, concerning matters about which there’s a clear majority view.
So if you’re a single guy looking for love, and you’re deciding which friend to bring out on the town with you, choose the one who’s pretty much exactly like you—only slightly less desirable.
“I must act not by what I think reasonable,” Condorcet wrote, “but by what all who, like me, have abstracted from their own opinion must regard as conforming to reason and truth.”
