Statistics 简明教程

Statistics - Probability Bayes Theorem

概率领域最重大的发展之一是贝叶斯决策理论的发展，这已被证明在不确定的条件下做出决策时具有巨大的帮助。贝叶斯定理是由英国数学家托马斯·贝叶斯牧师提出的。贝叶斯定理下的给定概率也被称为逆概率、后验概率或修正概率。这个定理通过考虑给定的样本信息来找到一个事件的概率，所以才叫后验概率。贝叶斯定理基于条件概率公式。

One of the most significant developments in the probability field has been the development of Bayesian decision theory which has proved to be of immense help in making decisions under uncertain conditions. The Bayes Theorem was developed by a British Mathematician Rev. Thomas Bayes. The probability given under Bayes theorem is also known by the name of inverse probability, posterior probability or revised probability. This theorem finds the probability of an event by considering the given sample information; hence the name posterior probability. The bayes theorem is based on the formula of conditional probability.

给定事件${B}$的事件${A_1}$的条件概率为

conditional probability of event ${A_1}$ given event ${B}$ is

同样，给定事件${B}$的事件${A_1}$的概率为

Similarly probability of event ${A_1}$ given event ${B}$ is

其中

Where

因此，贝叶斯定理的一般形式为

Hence the general form of Bayes Theorem is

其中${A_1}$, ${A_2}$…​${A_i}$…​${A_n}$是n个互相排斥且穷举的事件的集合。

Where ${A_1}$, ${A_2}$…​${A_i}$…​${A_n}$ are set of n mutually exclusive and exhaustive events.