-because half of the above quotations are fabricated by the author.

If human's understanding of vectors means grasping or even surpassing the limitation of time and space, then human's study of normal distribution is bound to symbolize the hope of controlling or even getting rid of fate.

Normal distribution, also known as Gaussian distribution, can be said to be one of the most important foundations in the scientific community. Its name comes from john carr Friedrich Goh, the prince of mathematics. ), that is, 10 years old, shota mentally calculated the sum of 1 and 100. (But it is not called orthographic distribution. )

Students of science and engineering are familiar with normal distribution. With the application of statistics in social sciences and even humanities (and the so-called "big data" craze), the application of normal distribution has broken through the traditional physical world and approached the spiritual world of human beings-from machine learning to neuroscience research, from behavioral psychological analysis to philosophical thinking.

Normal distribution has many elegant properties and wide applications, so I won't go into details here.

The most important one is called "Central Limit Theorem", which points out that when a large number of independent random variables are added together, their average values will show normal distribution.

Off-topic: Although everyone uses it in reality, it is almost impossible to achieve the two most fundamental conditions of being rich and independent of living things.

Advanced mathematics, calculus, advanced algebra, advanced geometry, probability statistics, etc. Where is the height?

Two words: limit.

This must be the greatest invention brought to mankind by the Enlightenment. Unfortunately, it is almost only used by the scientific community, so perhaps the word "discovery" is more accurate.

Hypergeometric distribution is also learning. Probability theory is one of the most basic distributions, but its application is far less extensive than normal distribution. In fact, its application is almost only related to sampling, such as the qualified rate of product sampling and the winning face of Texas Hold 'em.

Under certain conditions, it can also be approximated by normal distribution-essentially an application of the "central limit theorem" mentioned above.

The reason why the author relates hypergeometric distribution to human spiritual world is because of its application, or its definition is more inclined to the experimental thinking of probability theory from the beginning, rather than the observation and analysis of statistics.

Of course, the greatness of normal distribution lies in the realization of both.

Here, I want to repeat the reason why probability theory is more abstract than statistics.

The statistics are as follows: a fair coin is thrown 1000 times, and it is found that 46 1 time is face up; The frequency of frontal appearance is 0.461; If you throw 10000 times or more, the frequency will get closer and closer to 0.5.

Probability theory: for every fair coin, the probability of being upside down is 1/2.

In other words, probability theory is a priori and statistics is a posteriori.

By the way, it is Bayesian theorem that links the two.

No matter what the real world is like, the description of the physical world by human beings is often normally distributed.

When we estimate the distance, such as measuring the length of a table with a scale accurate to centimeters, we need to estimate the value of millimeters. When we shoot, throw and kick a ball at a target, such as shooting athletes who participate in the Olympic Games, everyone wants the gun to hit the bull's-eye ... Shooting percentage, gambling odds, per capita income and life expectancy all have the concept of expected value (that is, average value)-and this concept needs two conditions to be effective: 1 Probability distribution is only. 2. The probability distribution is almost symmetrical. In the qualified probability distribution, the normal distribution is the most convenient and lasting one.

In the physical world itself, due to the randomness of microscopic particles and a large number of macroscopic observations, at least when randomness is considered in classical physics, such as estimating and controlling rockets to fly into space, it is enough to use normal distribution to deal with the uncertainty in the whole process.

However, if the rocket is carrying an astronaut, he can unlock a panic button by entering a string of four-digit passwords (perhaps to launch a nuclear bomb to clear an inevitable asteroid)-and because of the tension, he only remembers the first two digits of the password, so he can only randomly enter the last two digits. He may make many mistakes and repeat them until he succeeds or dies.

In the whole process, the last password input and the previous password input are not independent of each other, and there are only 100 possibilities for a * *, and there is not much room for normal distribution to intervene.

Similarly, in the process of playing Texas Hold 'em, the probability that the next card is spades 10 with J, Q, K and A in hand has nothing to do with the normal distribution of half a dime.

The author thinks that hypergeometric distribution reflects the cognition of concepts such as strategic choice-they belong to the spiritual world and are characterized by being either one or the other in a few options.

The biggest difference between hypergeometric distribution and normal distribution is that it has a considerable skewness, especially when the sample base is not large.

There is a classic experiment: participants read the text of a mystery novel, list all the suspects in their eyes, and mark the probability that this person is the murderer.

The experimenter found that almost none of the subjects were on the suspect list, and the sum of the probabilities was100%-which was obviously illogical.

But is it really just because the subjects lack or ignore this basic statistical common sense?

When estimating the probability of a suspect, the subject is instinctively aware of the uncertainty of the value itself. In other words, this value itself is a random number, and it has its own probability function to describe its distribution.

When we are asked to write down only one numerical value to represent this function, we instinctively use the mode (corresponding to the peak value of the probability distribution) instead of the average value.

-this instinctive experience is that when we sample from samples that conform to this distribution, the number obtained is easier to approach the mode.

But the pattern is not average. For random numbers with symmetric probability distribution, mode, average and median are all one thing, but not for hypergeometric distribution.

"It is known that the sum of a set of random numbers is equal to 1, so the sum of their average values is also equal to 1. This proposition is undoubtedly correct, but the same proposition is invalid for most people.

In addition, for most subjects, "the probability that the suspect is the murderer" does not mean "the probability that the suspect is the murderer in this group", but "how confident I am that the suspect is the murderer".

Obviously, it's back to Bayesian.

In reality, too many research objects can't meet the requirements of using the central limit theorem at all, but it is abused after all-because it is convenient and easy to use, and the result will not be too bad in most cases.

Isn't it wonderful that many complex stochastic processes can be described by only one mean and one variance?

However, isn't it really because of laziness?

"Memory is unreliable, so the less you need to remember, the better." -this kind of nice words is just an excuse. The author joined the related major because he was too lazy to recite. Later, I found that I still had to recite ... and it was written in a language that I might not understand. )

But mathematicians are rigorous, and they will not abuse any theorems. Many times, they are so rigorous that many mathematical theories have conjectures before breakthroughs. Physicists have been pioneers for centuries, and computers have joined the trend since the Millennium.

Therefore, it is not mathematicians who abuse the theorem, but guys who use mathematical tools to do applied disciplines.

However, the author still blames mathematicians for the original sin of this kind of abuse, as if shirking responsibility.

Because they are so indifferent to applied disciplines, the most fundamental criticism and constructiveness of mathematics have not been passed down. Mathematics, as a compulsory course for all science and engineering majors, has not been widely loved, but has been openly despised, regardless of the East or the West.

-This must be a mathematician's problem.

Of course, the author can't understand that when a person is really immersed in what he loves, he doesn't need to compare with others and belittle others to get self-satisfaction.

From a single object to multiple objects, from certain events to random events, human beings gradually began to study their own behavior mathematically.

1994, John Forbes Nash Jr. and two other game theory experts won the Nobel Prize in Economics.

So why doesn't Nobel set up a math prize?

Because mathematicians always have other ways-this is a joke, but one of the most important conditions for awarding the Nobel Prize is "being alive".

Too many mathematicians died young, and it is rare that their mathematical theories can be widely used in their lifetime. Nash is definitely a lucky man.

As a graduate of Princeton University, Nash represents the direction of a new generation of mathematicians towards applied disciplines, especially economic-related fields.

Of course, the application of game theory goes far beyond economics, but in the end, economists all over the world are used to numbers today and begin to be superstitious about data. According to a statistic, graduates of Princeton economics department are more confident than those before and after World War II-because they use mathematical tools instead of analyzing problems based on historical experience.

It is not technology but faith that drives the evolution of human civilization. Network technology was mature before the arrival of the Internet era, but at that time, the Internet was only used by a small number of people and was not commercialized, so the whole world lacked infrastructure and related talents.

Computer technology has entered ordinary people's homes not because it is useful, but because its usefulness has been recognized by ordinary people's homes. Nash further promoted the mathematicization of economics after World War II, which in turn promoted the digitalization of financial institutions and general industries. More and more data are collected, and data analysis is becoming more and more useful. The whole world is connected by huge wires and optical cables, and the earth has really become a huge computer. The Hitchhiker's Guide to the Galaxy has become a prophecy.

On the other hand, if economics had not been mathematized by those generations, the numbers jumping in the financial market would be meaningless, and we would still live in an era when effective information can only be obtained through private channels-non-digital information is difficult to transmit through digital carriers. Remember that multimedia, such as music and video, came after the network developed sufficiently in recent years. Early computers and networks could only handle numbers and characters. Even so, the stock exchange is still the first place to adopt the Internet.

In a word, Nash's contribution is not only economic, but also economic.

2065438+In March 2006, AlphaGo defeated Li Shizhen.

Weiqi is a typical two-person zero-sum game with completely determined information.

The first thing I want to emphasize here is the difference between perfect information and complete information. To put it simply, under complete information, all the participants in the game know each other's goals; The complete information is only about the game itself.

For example, suppose a terrorist controls the global network system like in the movie and threatens the then South Korean president by launching a nuclear missile to attack South Korea, then Li Shizhen has to lose the game on purpose-and AlphaGo is just a Go AI, not a black hand behind the scenes; Then the game is not completely informative. But as long as the game is Weiqi and you can't change the position of the chess pieces by magic, then the game must be perfect.

The certainty is easy to understand, and there is no regret in moving. Chess pieces that want to move in the star position will not fall into small eyes or three or three for inexplicable reasons. Compared with randomness, certainty has two advantages.

First, for players, the game has a definite solution in theory.

Of course, in fact, the complexity of Go is very large, far exceeding all kinds of chess. At present, the most skilled chess player is chess AI, and there is a professional game of "chess player +AI" team playing. AI gives advice to players, and players can freely choose whether to follow it or not, or find another way. Although most of the time, the advice given by AI will be favored, there are also many players who have an epiphany and walked out of the moves that two AI did not expect to win.

The complexity of Weiqi has nothing to do with the way the chess pieces move, but entirely stems from its huge 19x 19 chessboard. Just as we can use a small chessboard in the early days of learning Go, the early Go AI also started from challenging the small chessboard.

Second, there is a clear distinction between black and white. If you win, you win, and if you lose, you lose. The match between AlphaGo and Li Shizhen is five games, that is, two wins in three games. If this game is a coin toss, anyone can beat AI and anyone can win the world championship.

Most importantly, because the benefits of the game (whether name or profit) are directly linked to the results on the chessboard, there is almost no difference between perfect information and complete information. Even if Li Shizhen really wants to lose, the fact that he lost the game will not change.

Finally, Go is a one-on-one zero-sum game.

In the case of war games (or the real war itself), this game is zero-sum, so direct fighting is inevitable. But if there are many troops, it is entirely possible to form an alliance to invade and annihilate the enemy.

The United States and the Soviet Union have always had their own purposes, but in order to defeat the Nazis, they finally United, although only temporarily. But even at that time and place, they can't really cooperate. It is said that wily politicians must calculate the distribution of benefits after the war before it begins. If neither side has such leadership, how can a cold war pattern with almost symmetrical strength be formed after the war?

However, random games can bring problems.

It is generally believed that the problem of randomness lies in the composition of luck. In other words, if Go is a game of random elements, and AlphaGo wins three out of five games, humans can shout "bad luck" and then ask for another 300 rounds.

But this problem has been solved under this stubborn request-that is, through many games, we can see the average score.

In 20 15, Professor Michael Bowling of university of alberta published a paper in Science magazine: He and his colleagues "weakly" solved the Double Texas Hold 'em-the program Cepheus they developed, which can be guaranteed to be unbeaten when they know that they have cards on both sides. Note that the "weakness" here emphasizes knowing the cards of both sides, that is, simplifying Texas Hold 'em, which was originally a game with incomplete information, into a game with complete information; And "unbeaten" doesn't mean that you won't lose every game, but from many games, if you gamble with Cepheus, you won't lose on average.

Of course, the question in reality is, how many times does it take to prove which is better and which is worse? Both humans and AI participating in the competition will change, so strictly speaking, there is no way to conduct completely repeated experiments.

The next problem is incomplete information.

In a sense, incomplete information can also be random. Continuing to take poker as an example, spades A is certain about himself, but uncertain about his opponent.

However, the impact of this uncertainty on the strategy is quite different from the randomness of the mutual agreement mentioned in the previous paragraph. Although for opponents, the probability of "this unknown card is spades after all" is the same as that of "Zhang Xinfa's next card is spades"; However, the probability of "A of spades raises based on this card" is completely different from that of "A of spades raises based on this card". The uncertainty of the card itself combines the uncertainty of all possible strategies based on the card, which is the real uncertainty now.

In the case of regression theory, such uncertainty only increases the amount of calculation-but there is no doubt that AI cracks random games, so it becomes more difficult.

Finally, the question of multiplayer games.

On the premise of imperfect information game, multiplayer games first add unknown cards, which directly increases the uncertainty of quantity.

At the same time, as mentioned in the previous section, it is possible for multiplayer games to form alliances in reality, and the parties to the game may not realize the existence of similar agreements-that is, multiplayer games aggravate the imperfection of information and compound multi-level uncertainty again in structure.

The most critical question is, how do we test who is better than human beings in a certain (or this kind of) AI? Suppose a team develops an excellent mahjong AI, does it make sense for two AIs to compete with two humans? Will humans communicate with each other for human dignity? Does AI cheat, too?

It may be reasonable to set up such an AI on the online game platform: players entering the game cannot judge whether their opponents are AI according to their ID; Enough games. -but in order to win, it is what professional players should do to study the opponent's card path (chess path) in advance. Is such anonymity inherently unfair to human nature? For example, AlphaGo has collected many classic chess manuals, including Li Shizhen himself, but AlphaGo's own chess manuals are pitiful. Li Shizhen was already at a disadvantage before the game started.

In the process of introducing random multiplayer games with incomplete information, the author gives two typical examples: poker and mahjong.

The author's question is: Why is this game always directly related to gambling?

This question seems a bit unreasonable. By definition, almost all sports are random games.

For example, shooting, excellent players can make every shot close to the bull's-eye, but it is impossible to guarantee that every shot hits the bull's-eye, so the game is not a duel, but a shot decides the outcome. For example, in track events, the results will definitely be affected by the physical condition and weather conditions of the players, especially the sprint and "gun-pressing tactics", which add a lot of highlights to the competition. Team sports, such as football, are influenced by too many random factors, either natural or artificial.

It's not that sports have nothing to do with gambling, such as gambling. But as long as the athlete himself doesn't lose the bet, he will try to win in the end, that is to say, in principle, the sport can be independent of gambling and only fight for honor. However, even if Texas Hold 'em and Mahjong are all gambling, after all, props such as chips should be used to express the winning or losing amount.

In addition, imperfect information is not a problem, which is very common in team events: a series of fancy moves given by hard balls, a gesture of a volleyball setter gesturing behind his teammates when serving, and a small note that Lyman got before the penalty shoot-out of the 200614 World Cup. ...

As for multiplayer events, there seems to be no single event like poker or mahjong in sports competitions. For example, F 1, long-distance running and cycling all emphasize teamwork. Armstrong won the Tour de France seven times in a row, not just because he took drugs. However, similar cooperation can easily be regarded as cheating in poker or mahjong. I won't discuss the difference of this concept here, because it is likely that gambling comes first, which gives birth to the taboo of teamwork, not the other way around.

Based on China and Japan, the worldwide mahjong competition is progressing slowly.

Relevant domestic institutions are implementing a rule similar to "repeated bridge", that is, the composition of each game at different tables is the same as that of four players in the southeast, northwest and northwest. Like bridge, the purpose is naturally to reduce randomness.

However, I don't think it is meaningful-or counterproductive-to reduce randomness by increasing the relevance of players at different tables. The number of randomly generated cards is reduced, which actually reduces the sample size and increases the variance.

Of course, because the game is completely played on the computer, and the card effect of the player is also recorded by the computer, it is still of great significance.

In order to reduce randomness, Japan's top competitions adopt "competition", also called "one rule", which is not much different from the general Japanese hemp, mainly reducing the number of precious cards.

Speaking of reducing randomness, vulture nest mahjong is probably a good way:

Of course, the last one is unnecessary.

When the author claims that the movement is random, I believe many people agree, but they don't want to agree, at least they don't want to agree completely.

As long as you constantly improve your level, you will eventually crush weak opponents. But this kind of competition is not what the audience or even the top athletes yearn for. From the bottom of our hearts, we are eager for a peak confrontation in history.

Yes, this is the charm of competition: two evenly matched masters, Jian and Ji, threatened to kill themselves, for a moment, a fraction of a second. ...

And games like poker or mahjong, which depend entirely on luck, even beginners may beat the world champion.

If it's just one game, that's for sure, but what about the whole game? Top competitions, especially the championship, last more than a whole day, or even two days? -In this case, if the novice can win the championship, isn't it really the problem of other players?

The vast torrent of the whole, supplemented by the uncertainty of the details, is our mentality.

Our ancestors escaped the attack of wild animals and survived the turmoil. In the ancient times of reproductive worship, there were more goodness, more sacredness and more hymns.

However, I don't know when "scarcity is the most valuable thing" is not only the natural balance of supply and demand, but also directly printed in our culture and genes.

We give this rule to all the games we invented, such as war, hunting, sports and chess-the hardest to get and the most valuable.

What is the biggest difference between chess and other examples?

-that is, the rules of chess and cards are formulated by us and are not bound by physical laws.

We are used to the normal distribution of the material world, but we are just little urchins in the hypergeometric distribution of the spiritual world. For unconstrained people, the methods of discipline have been:

Gambling is the best way to make the participants take responsibility for winning or losing the material data.

After answering the questions at the beginning of this chapter, I will finally talk about two distributions, or the difference between the two worlds.

In the physical world, we struggle with the negative correlation between mean and variance. Greater returns often mean expensive costs and high risks.

This is not to say that this is not the case in the rules of the game we have formulated. It is also the author's extreme to directly correspond the hypergeometric objects to the spiritual world.

But normal distribution is based on things themselves, while hypergeometric distribution is based on combination, that is, the relationship between things-such a generalization should be pertinent enough.

The object of the material world, as well as our general experience, comes from the repetition and accumulation of things. However, since the birth of living things, or earlier, since the appearance of biological macromolecules, the complexity of evolution is not only quantitative, but also structural. And human society is the so-called "superstructure"-just as living things, as a physical existence, must follow all physical laws, but there is no need to study the coat color of cats and dogs with quantum mechanics and relativity-our spiritual world is based on the physical world, which does not mean that the laws of the two worlds are universal.

Finally, I want to extend my most sincere respect and deepest regret to the whole enlightenment era.

The Enlightenment advocated rationality-not only in scientific research, but also in the self-discovery of human nature and the whole social civilization-but everything is just an illusion, and human beings will only take responsibility in the oldest way.