Gaussian Quotes

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics.

It means this War was never political at all, the politics was all theatre, all just to keep the people distracted…secretly, it was being dictated instead by the needs of technology…by a conspiracy between human beings and techniques, by something that needed the energy-burst of war, crying, “Money be damned, the very life of [insert name of Nation] is at stake,” but meaning, most likely, dawn is nearly here, I need my night’s blood, my funding, funding, ahh more, more…The real crises were crises of allocation and priority, not among firms—it was only staged to look that way—but among the different Technologies, Plastics, Electronics, Aircraft, and their needs which are understood only by the ruling elite… Yes but Technology only responds (how often this argument has been iterated, dogged, humorless as a Gaussian reduction, among the younger Schwarzkommando especially), “All very well to talk about having a monster by the tail, but do you think we’d’ve had the Rocket if someone, some specific somebody with a name and a penis hadn’t wanted to chuck a ton of Amatol 300 miles and blow up a block full of civilians? Go ahead, capitalize the T on technology, deify it if it’ll make you feel less responsible—but it puts you in with the neutered, brother, in with the eunuchs keeping the harem of our stolen Earth for the numb and joyless hardons of human sultans, human elite with no right at all to be where they are—” We have to look for power sources here, and distribution networks we were never taught, routes of power our teachers never imagined, or were encouraged to avoid…we have to find meters whose scales are unknown in the world, draw our own schematics, getting feedback, making connections, reducing the error, trying to learn the real function…zeroing in on what incalculable plot? Up here, on the surface, coal-tars, hydrogenation, synthesis were always phony, dummy functions to hide the real, the planetary mission yes perhaps centuries in the unrolling…this ruinous plant, waiting for its Kabbalists and new alchemists to discover the Key, teach the mysteries to others…

Many of the most interesting phenomena that we have touched upon fall into this category, including the occurrence of disasters such as earthquakes, financial market crashes, and forest fires. All of these have fat-tail distributions with many more rare events, such as enormous earthquakes, large market crashes, and raging forest fires, than would have been predicted by assuming that they were random events following a classic Gaussian distribution.

I can remember one occasion, taking a shower with my wife while high, in which I had an idea on the origins and invalidities of racism in terms of Gaussian distribution curves. It was a point obvious in a way, but rarely talked about. I drew curves in soap on the shower wall, and went to write the idea down. One idea led to another, and at the end of about an hour of extremely hard work I had found I had written eleven short essays on a wide range of social, political, philosophical, and human biological topics. . . . I have used them in university commencement addresses, public lectures, and in my books.

One of the problems I face in life is that whenever I tell people that the Gaussian bell curve is not ubiquitous in real life, only in the minds of statisticians, they require me to “prove it”—which is easy to do, as we will see in the next two chapters, yet nobody has managed to prove the opposite

Before we move on, let me clarify that there is a fundamental difference between what we do and how predictable we are. When it comes to things we do-like the distances we travel, the number of e-mails we send, or the number of calls we make-we encounter power laws, which means that some individuals are significantly more active than others. They send more messages; they travel farther. This also means that out-liers are normal-we expect to have a few individuals, like Hasan, who cover hundreds or even thousands of miles on a regular basis. But when it comes to the predictability of our actions, to our surprise power laws are replaced by Gaussians. This means that whether you limit your life to a two-mile neighborhood or drive dozens of miles each day, take a fast train to work or even commute via airplane, you are just as predictable as everyone else. And once Gaussians dominate the problem, outliers are forbidden, just as bursts are never found in Poisson's dice-driven universe. Or two-mile-tall folks ambling down the street are unheard of. Despite the many differences between us, when it came to our whereabouts we are all equally predictable, and the unforgiving law of statistics forbids the existence of individuals who somehow buck this trend.

But you understand the relationship between pi and Gaussian curvature, right?

FIGURE 5.8: Three assumptions of the linear model (left side): Gaussian distribution of the outcome given the features, additivity (= no interactions) and linear relationship. Reality usually does not adhere to those assumptions (right side): Outcomes might have non-Gaussian distributions, features might interact and the relationship might be nonlinear.

Alice: "Before the comparison with the threshold, I apply a Fisher transformation. This gives the curve a Gaussian distribution with sharp and well defined oscillations, so we get less false signals." Bob: "I have no idea what you're talking about.

Assuming the total error was approximately normally distributed (the Gaussian or bell-shaped curve), we needed the standard deviation (a measure of uncertainty) for the error of prediction around the actual outcome to be sixteen pockets (0.42 revolution) or less to get an edge. We achieved the tighter estimate of ten pockets, or 0.26 revolution. This gave us the enormous average profit of 44 percent of the amount we bet on the forecast number. If we spread our bet over the two closest numbers on each side, for a total of five numbers in all, we cut risk and still had a 43 percent advantage.

Noll tried to register Gaussian Quadratic with the US Copyright Office at the Library of Congress, another body perplexed by the works on display. His request was originally denied “since a machine had generated the work.”10 He explained that a human being had written the program that, through a mix of randomness and order, generated the work. The Library of Congress again declined: randomness was unacceptable. Noll finally argued that although the numbers produced by the program appeared random, “the algorithm generating them was perfectly mathematical and not random at all,” and the work was finally patented.

Every now and then, more often than in the boring world of the Gaussian distribution, events from the Pareto distribution arise to shake our world.

In the GRW scheme, however, an object as large as a cat, which would involve some 10^27 nuclear particles, would almost instantaneously have one of its particles 'hit' by a Gaussian function (as in Fig. 6.2), and since this particle's state would be entangled with the other particles in the cat, the reduction of that particle would 'drag' the others with it, causing the entire cat to find itself in the state of either life or death. In this way, the X-mystery of Schrodinger's cat-and of the measurement problem in general-is resolved.

there are only four units in the ring R-1 of Gaussian integers, namely ±1 and ±i; multiplication by any of these units effects a symmetry of the infinite square tiling

What I am telling you here is actually nothing new. So why switch from analyzing assumption-based, transparent models to analyzing assumption-free black box models? Because making all these assumptions is problematic: They are usually wrong (unless you believe that most of the world follows a Gaussian distribution), difficult to check, very inflexible and hard to automate. In many domains, assumption-based models typically have a worse predictive performance on untouched test data than black box machine learning models. This is only true for big datasets, since interpretable models with good assumptions often perform better with small datasets than black box models. The black box machine learning approach requires a lot of data to work well. With the digitization of everything, we will have ever bigger datasets and therefore the approach of machine learning becomes more attractive. We do not make assumptions, we approximate reality as close as possible (while avoiding overfitting of the training data).

first central assumption leading to the Gaussian bell curve fails in reality. In games, of course, past winnings are not supposed to translate into an increased probability of future gains—but not so in real life, which is why I worry about teaching probability from games. But when winning leads to more winning, you are far more likely to see forty wins in a row than with a proto-Gaussian.

If I look at it with a magnifying glass, the terrain will be smoother but still highly uneven. But when I look at it from a standing position, it appears uniform—it is almost as smooth as a sheet of paper. The rug at eye level corresponds to Mediocristan and the law of large numbers: I am seeing the sum of undulations, and these iron out. This is like Gaussian randomness: the reason my cup of coffee does not jump is that the sum of all of its moving particles becomes smooth. Likewise, you reach certainties by adding up small Gaussian uncertainties: this is the law of large numbers.

The profit distribution of real traders is not a Gaussian, but a Lévy distribution. It has a smaller peak and fatter tails. That means the losers lose more, and the winners take more than in a random-trading situation.

One of the most misunderstood aspects of a Gaussian is its fragility and vulnerability in the estimation of tail events. The odds of a 4 sigma move are twice that of a 4.15 sigma. The odds of a 20 sigma are a trillion times higher than those of a 21 sigma! It means that a small measurement error of the sigma will lead to a massive underestimation of the probability. We can be a trillion times wrong about some events.

to stand for “periphery.” It is hard to ignore the ubiquity of pi in nature. Pi is obvious in the disks of the moon and the sun. The double helix of DNA revolves around pi. Pi hides in the rainbow and sits in the pupil of the eye, and when a raindrop falls into water, pi emerges in the spreading rings. Pi can be found in waves and spectra of all kinds, and therefore pi occurs in colors and music, in earthquakes, in surf. Pi is everywhere in superstrings, the hypothetical loops of energy that may vibrate in many dimensions, forming the essence of matter. Pi occurs naturally in tables of death, in what is known as a Gaussian distribution of deaths in a population. That is, when a person dies, the event “feels” the Ludolphian number. It is one of the great mysteries why nature seems to know mathematics.

three essential components: single-task neural networks, multi-task neural networks, and Gaussian process regression.

We used Matlab code released by Carl Rassmussen and Chris Williams to accompany their Gaussian processes book.