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June 5 - July 3, 2021
set some variables to zero.
mathematician is always asking, “What assumptions are you making? And are they justified?”
Nonlinear thinking means which way you should go depends on where you already are.
method of exhaustion.
This can also be applied to valuing businesses (some sort of triangulation)
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.
The slogan to keep in mind: straight locally, curved globally.
Newton said, look, let’s go all the way. Reduce your field of view until it’s infinitesimal—so small that it’s smaller than any size you can name, but not zero. You’re studying the missile’s arc, not over a very short time interval, but at a single moment. What was almost a line becomes exactly a line. And the slope of this line is what Newton called the fluxion, and what we’d now call the derivative.
So −T= T − 1, an equation concerning T which is satisfied only when T is equal to 1/2. Can a sum of infinitely many whole numbers somehow magically become a fraction? If you say no, you have the right to be at least a little suspicious of slick arguments like this one. But note that some people said yes, including the Italian mathematician/priest Guido Grandi, after whom the series 1 − 1 + 1 − 1 + 1 − 1 + . . . is usually named; in a 1703 paper, he argued that the sum of the series is 1/2, and moreover that this miraculous conclusion represented the creation of the universe from nothing.
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.
William Carlos Williams put it crisply: no ideas but in things.
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.
The Law of Large Numbers will always push the Big players’ scores toward 50%, while those of the Smalls are apt to vary much more widely.
The smaller the number of coins—what we’d call in statistics the sample size—the greater the variation in the proportion of heads.
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.
Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.
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 point of Bennett’s paper is to warn that the standard methods of assessing results, the way we draw our thresholds between a real phenomenon and random static, come under dangerous pressure in this era of massive data sets, effortlessly obtained. We need to think very carefully about whether our standards for evidence are strict enough, if the empathetic salmon makes the cut.
didn’t use statistical safeguards (known as “multiple comparisons correction”) that take into account the ubiquity of the improbable.
Improbability, as described here, is a relative notion, not an absolute one; when we say an outcome is improbable, we are always saying, explicitly or not, that it is improbable under some set of hypotheses we’ve made about the underlying mechanisms of the world.
It’s not enough that the data be consistent with your theory; they have to be inconsistent with the negation of your theory, the dreaded null hypothesis.
This is important
Assuming the truth of something we quietly believe to be false is a time-honored method of argument that goes all the way back to Aristotle; it is the proof by contradiction, or reductio ad absurdum.
impossible and improbable are not the same—not even close. Impossible things never happen. But improbable things happen a lot.
Among the first N numbers, about N/log N are prime; this is the Prime Number Theorem,
The Prime Number Theorem says that, among the first N integers, a proportion of about 1/log N of them are prime.
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.
For Neyman and Pearson, the purpose of statistics isn’t to tell us what to believe, but to tell us what to do. Statistics is about making decisions, not answering questions.
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.
In 2013, the Association for Psychological Science announced that they would start publishing a new genre of article, called Registered Replication Reports. These reports, aimed at reproducing the effects reported in widely cited studies, are treated differently from usual papers in a crucial way: the proposed experiment is accepted for publication before the study is carried out. If the outcomes support the initial finding, great news, but if not, they’re published anyway, so the whole community can know the full state of the evidence.
There’s a hard limit to how far in advance we can predict the weather, no matter how much data we collect.
Due to chaos theory
“The chance that the null hypothesis is correct, given that we observed a certain experimental result.”
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.
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.
“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.”
The old saying tells us lotteries are a “tax on the stupid,”
A better name might be “average value”—for what the expected value of the bet really measures is what I’d expect to happen if I made many such bets on many such dogs.
Instead of expected value
fact not obvious at all. If they were, they would not have arrived so late in the history of human thought.
J / 175 million + 36.7 cents.
additivity of expected value.
E(X+Y)= E(X) + E(Y).
George Stigler, the 1982 Nobelist in economics, used to say, “If you never miss the plane, you’re spending too much time in airports.”
decisions must be made, and economists aspire to tell us how to make them,
“The interest I have to believe a thing is no proof that such a thing exists.” Voltaire
it was possible to make mathematical calculations agree with common sense without introducing some moral considerations;
Pierre-Simon Laplace, on the last page of his 1814 treatise A Philosophical Essay on Probabilities, writes, “We see, in this Essay, that the theory of probabilities is, in the end, only common sense boiled down to ‘calculus’; it points out in a precise way what rational minds understand by means of a sort of instinct, without necessarily being aware of it. It leaves nothing to doubt, in the choice of opinions and decisions; by its use one can always determine the most advantageous choice.”
As Donald Rumsfeld would later put it, there are known unknowns and there are unknown unknowns, and the two are to be processed differently.
In the decision-theory literature, the former kind of unknown is called risk, the latter uncertainty.
Battling variance is one of the main challenges of managing money, whether you call it that or not.
Abu ‘Ali al-Hasan ibn al-Haytham (but let’s call him Alhazen, as most Western writers do). His treatise on optics, the Kitab al-Manazir, was translated into Latin and taken up eagerly by philosophers and artists seeking a more systematic understanding of the relation between sight and the thing seen.
