Our brains will also produce misconceptions and blind spots, just as drivers have "blind spots" in their eyesight, which can only be eliminated by looking at the bathroom mirror several times, so their thinking process also has blind spots, and they must respond through calculation and thinking. Probability theory is an industry that often draws strange conclusions contrary to judgment. Even mathematicians will make big mistakes if they are not careful. Now, let's first give an example of a fallacy in traditional probability, which is called "basic proportion".
From an example in life, we gradually went to the hospital for an examination to check the probability that he got some kind of disease. The result turned out to be positive, which shocked her and quickly inquired online. According to the information on the Internet, the test is always biased. This test has "false positive rate 1% and false negative rate 1%". The meaning of this sentence is that 1% is false negative and 99% is true positive. However, 1% is a false positive and 99% is a true negative among people who are not sick. So according to this statement, wang hong himself may have a 99% chance of suffering from this disease.
Wang Hong thinks that even if the false positive rate is only 1% and 99% are true positive, then the probability that my population has been infected with this disease should be 99%. But the doctor told him that the probability of her being infected in the general population was only about 0.09(9%). What's going on here? Where is wang hong's wrong idea?
Color gallery: Dr. pexls said, "99%? There is not such a great sense of probability. 99% is the accuracy of the test, not the probability that you are sick. You forgot one thing: the normal infection rate of this disease is not large, and only one person in 1000 people is sick. " Originally, this doctor loves to study mathematics in addition to medicine, and probability method is often used in medicine.
Most of his calculation methods are as follows: due to the missing rate of detection is 1%, soon 1000 people will be reported as "false positive", according to the proportion of this disease in the population (11000 = 0.1%). 1 1 Only one of the positive people is really positive (sick), so the probability of wang hong being infected is probably11/,which is 0.09(9%).
Wang Hongsi still feels confused after thinking about it, but this incident has strengthened Wang Hong's memory of the probability theory he learned before. After reading the article and thinking about the doctor's optimization algorithm, he realized that he had made a mistake, called "fallacy of basic proportion", that is, he forgot to apply the fact that "the most basic proportion of this disease in the population is (11000)".
When it comes to the fallacy of proportion, we'd better start with the famous Bayes theorem in probability theory. Thomas Bayes (1701-1761) is a British statistician and used to be a mage. Bayesian theorem is his great contribution to probability theory and applied statistics, and it is the basic framework of machine learning algorithm commonly used in artificial intelligence technology today. Its concept is far more profound than ordinary people's cognitive ability. Perhaps Bayesian himself didn't know it before his death. Because of such a key achievement, he did not publish it before his death, and it was published by his good friend after his death 1763.
Roughly speaking, Bayesian theorem involves the interaction between two random variables A and B. If summarized in one sentence, these laws are about how to modify the "prior probability" P(A) of A when B does not exist, so as to obtain the "standard probability" P(A|B) after B exists, or a posteriori probability. If you write a formula to calculate:
The definitions of a priori and a posteriori here are conventional and relative. For example, A and B can also be described in reverse, that is, how to get the "standard probability" P(B | a) of B from the prior probability p (b) of B, as shown by the diagonal line in the figure.
Don't be afraid of formula calculation. According to examples, we can gradually understand it. For example, in the case of wang hong's medical treatment, the random variable A means "Wang Hongde has some kind of disease"; The random variable b represents "the test result of wang hong". The prior probability P(A) refers to the probability that Wang Hong got the disease without any test results (that is, the most basic probability of the disease in the public is 0.1%); The standard probability (or a posteriori probability) P(A|B) refers to the probability (9%) that Wang Hong gets the disease when the test result is positive. How to adjust from basic probability to posterior probability? We'll talk later.
Bayesian theorem is the product of 18 new century. It will be easy to use next year in 2000, and I don't want to be tested in the 1970s. This test comes from the "basic proportion fallacy" proposed by DanielKahneman and Tversky. The former is an African-American psychologist who won the Nobel Prize in Economics in 2002.
The fallacy of proportion is basically not to deny Bayesian theorem, but to discuss a puzzling question: why does human intuition often run counter to the value of Bayesian formula? As the example just showed, people often ignore the basic probability when using judgment. Kahneman and others disdain a taxi in their article Thinking, Fast and Slow, in order to inspire everyone to consider the main reasons that endanger everyone's "management decision-making". We don't want to talk about the value of the basic proportional fallacy to "decision theory" here, but only use this example to enhance our understanding of Bayesian formula.
Suppose a city has blue and green taxis (market share is 15∶85). A taxi hit and run at night, but fortunately there were witnesses. Witnesses estimated that the taxi of the perpetrator was blue. However, what about his "authenticity"?
The public security organs conducted a "green and blue" test on witnesses in the same natural environment, and concluded that 80% of the cases were correctly identified and 20% were wrongly identified. Perhaps some readers got the result immediately: the probability that the accident car is blue should be 80%. If you make such a response, you are making the same mistake as wang hong in the above example, ignoring the prior probability and not considering the most basic "green and blue" car ratio in this city.