Bayes' theorem was developed by the English mathematician Thomas Bayes (1702-1761) to describe the relationship between two conditional probabilities , such as P(A|B) and P(B|A). According to the multiplication rule, it can be immediately derived: P(A∩B) = P(A)*P(B|A)=P(B)*P(A|B). The above formula can also be transformed into: P
definition
edit
There is a basic tool in Bayesian
statistics called Bayesian formula, also known as Bayesian rule. Although it is a mathematical formula, its principle can be understood without numbers. If you see a person doing something good all the time, that person is probably a good person. That is to say, when you cannot accurately know the essence of a thing, you can rely on the occurrence of events related to the specific essence of the thing to judge the
probability
. Expressed in mathematical language: the more events that support a property, the more likely it is to hold.
Bayesian formula is also known as Bayesian theorem, and Bayesian rule is a standard method for correcting subjective judgments (i.e. prior probability) about
probability .
The so-called Bayesian formula means that when the analysis sample is large enough to be close to the population, the probability of events in the sample will be close to the probability of events in the population. However,
behavioral
economists have found that people often do not follow Bayesian law in the decision-making process, but give more weight to recent events and the latest experience, and place too much importance on recent events when making decisions and making judgments. event. Faced with complex and general problems, people often take shortcuts and make decisions based on probabilities rather than probabilities. This systematic deviation from the classical model is called "bias". Due to the existence of psychological biases, investors are not absolutely rational when making decisions and judgments, and will behave biased, which in turn
affects
the price changes in the capital market. But for a long time,
economists
have had to stick to Bayes' rule in their analysis because of the lack of powerful alternative tools.
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principle
editBayes' rule
Usually, the probability of event A under the condition of event B (occurrence) is not the same as the probability of event B under the condition of event A; however, there is a definite relationship between the two, and Bayes' rule is such statement of relationship.
As a canonical principle,
Bayes' rule
is valid for all interpretations of probability; however, frequentists and Bayesians have different views on how probabilities are
assigned
: frequentists are based on the occurrence of random events The frequency of , or the number in the overall sample to assign probabilities; Bayesians assign probabilities based on unknown propositions. As a result, Bayesians have more opportunities to use Bayes' rule.
where P(A|B) is the probability that A occurs if B occurs.
is a complete event group, i.e.
在
贝叶斯法则中，每个名词都有约定俗成的名称：
Pr(A)是A的
先验概率或边缘概率。之所以称为"先验"是因为它不考虑任何B方面的因素。
Pr(B|A)是已知A发生后B的条件概率，也由于得自A的取值而被称作B的后验概率。
Pr(B)是B的先验概率或边缘概率，也作标准化常量（normalized constant）。
按这些术语，Bayes法则可表述为：
后验概率 = (似然度 * 先验概率)/标准化常量 也就是说，后验概率与先验概率和似然度的乘积成正比。
另外，比例Pr(B|A)/Pr(B)也有时被称作标准似然度（standardised likelihood），Bayes法则可表述为：
贝式定理
对于变量有二个以上的情况，贝式定理亦成立。例如：
P(A|B,C)=P(B|A)*P(A)*P(C|A,B)/(P(B)*P(C|B))
意义
编辑
贝叶斯定理公式
(3张)
我们假设 A 事件为狗在晚上叫，B 为盗贼入侵，则以天为单位统计，P(A) = 3/7，P(B) = 2/(20*365) = 2/7300，P(A|B) = 0.9，按照公式很容易得出结果：P(B|A) = 0.9*(2/7300) / (3/7) = 0.00058
另一个例子，现分别有 A、B 两个容器，在容器 A 里分别有 7 个红球和 3 个白球，在容器 B 里有 1 个红球和 9 个白球，现已知从这两个容器里任意抽出了一个球，问这个球是红球且来自容器 A 的概率是多少?
假设已经抽出红球为事件 B，选中容器 A 为事件 A，则有：P(B) = 8/20，P(A) = 1/2，P(B|A) = 7/10，按照公式，则有：P(A|B) = (7/10)*(1/2) / (8/20) = 0.875
贝叶斯公式为利用搜集到的信息对原有判断进行修正提供了有效手段。在采样之前，经济主体对各种假设有一个判断（
先验概率），关于先验概率的分布，通常可根据经济主体的经验判断确定（当无任何信息时，一般假设各先验概率相同），较复杂精确的可利用包括最大熵技术或边际分布密度以及相互信息原理等方法来确定先验
概率分布。
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