贝叶斯定理

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During World War II, cryptoanalysts in both the United States and the UK were frantically trying to decipher encrypted communications sent by the Axis powers’ military branches. Among these efforts, the British code-breaking center at Bletchley Park probably stands out in its fame as the birthplace of the British Bombe, the machine that would decode the infamous German cipher Enigma between 1939 and the end of the war. While the Bombe and its successor models have received most of the public’s attention, their efforts would often have been futile without the support of statistical methods. The goal of this article is to cast light on one of these methods, namely Bayesian inference, first by introducing the theory behind it and then by outlining how it was used to crack Axis ciphers.

第二次世界大战期间，美国和英国的加密分析师都在疯狂地试图破译由轴心国军事分支发送的加密通信。在这些努力中，位于布莱奇利公园(Bletchley Park)的英国密码破解中心可能以英国邦贝(Bombbe)的发源地而出名，该机器可解码1939年至战争结束之间臭名昭著的德国密码《谜》。尽管孟买及其后继模型受到了大多数公众的关注，但如果没有统计方法的支持，他们的努力往往是徒劳的。本文的目的是首先介绍一种方法，即贝叶斯推理，首先介绍其背后的理论，然后概述其如何用于破解Axis密码。

Bletchley Park, photographed by Toby Oxborrow. Source: Flickr. This file is licensed under the 布莱奇利公园的小屋1，由Toby Oxborrow拍摄。资料来源：Flickr。此文件已获得Creative Commons 知识共享Attribution-Share Alike 2.0 Generic license署名-相同方式共享2.0通用许可的许可

Part I will be dedicated to the breaking of the Japanese Naval cipher JN 25, whereas Part II will outline the role Bayesian inference played in cracking Enigma. Large parts of this report are based on an article written by Edward Simpson, one of the code-breakers who worked at Bletchley Park. I have tried to make the topic more accessible by reviewing the Bayesian formalism and carrying out most of the math that is only implied in Simpson’s article. If at the end of this post, you find yourself intrigued by the topic, I encourage you to read the original article by Simpson for a fascinating first-hand report of his time at Bletchley Park.

第一部分将致力于打破日本海军密码JN 25，而第二部分将概述贝叶斯推理在破解《谜》中的作用。该报告的大部分内容基于爱德华·辛普森(Edward Simpson)撰写的一篇文章，该文章是在Bletchley Park工作的代码破坏者之一。我试图通过回顾贝叶斯形式主义并执行仅在Simpson文章中隐含的大部分数学运算，来使该主题更易于访问。如果在本文结尾处，您发现自己对该主题感兴趣，我鼓励您阅读辛普森(Simpson)的原始文章，以获得有关他在布莱奇利公园(Bletchley Park)的时间的有趣的第一手报道。

定理 (The theorem)

Like so many other famous theories, Bayes’ theorem is surprisingly simple:

像许多其他著名理论一样，贝叶斯定理非常简单：

The formula itself is easily derived and has many applications outside of Bayesian statistics. However, its simplicity can be deceiving as most of the theorem’s power lies in the interpretation of the probabilities P involved. The true controversy that has stuck with Bayes’ theorem for centuries lies in the way its usage challenges the more traditional frequentist approach. While so-called frequentists define the probability of an event as the “limit of its relative frequency in many trials” [1]. Bayesians interpret probability as a measure of personal belief.

该公式本身易于导出，并且在贝叶斯统计之外具有许多应用。但是，它的简单性可能是骗人的，因为大多数定理的能力在于对涉及的概率P的解释。几个世纪以来一直存在于贝叶斯定理的真正争议在于它的用法挑战了更传统的常客主义方法。所谓的常客主义者将事件的可能性定义为“在许多试验中其相对频率的极限” [1]。贝叶斯主义者将概率解释为个人信念的量度。

One may ask how they dare include something as subjective as belief in a mathematical theory. And voicing this critique, one would certainly be in good company. Many heavyweights of statistics, including Fisher and Pearson, have discarded the Bayesian interpretation of probability based on similar arguments [4].

有人可能会问，他们怎么敢在数学理论中包含主观信念。说出这种批评，肯定会有好伙伴。许多统计重量级人物，包括Fisher和Pearson，都基于相似的论点放弃了对概率的贝叶斯解释[4]。

To really understand the difference between the two approaches and be able to pass fair judgment, let us consider the following example:

要真正理解两种方法之间的区别并能够通过公正的判断，让我们考虑以下示例：

Imagine a friend challenges you to a game of flipping coins and promptly produces a coin that she would like to use. Before agreeing to this game you, being rather suspicious by nature, would like to ascertain that the coin she handed you is fair, i.e. the chances of it landing on head and tail are equal.

想象一个朋友向您挑战掷硬币游戏，并立即产生她想使用的硬币。 在同意此游戏之前，您本性相当可疑，所以想确定她递给您的硬币是否公平，即它落在头和尾上的机会是相等的。

惯常做法 (Frequentist approach)

A frequentist will frame this problem as a hypothesis test with null hypotheses H₀: the coin is fair, and alternative hypotheses H₁ that it is not. She will then decide on a number of trials (say,

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贝叶斯定理_贝叶斯定理如何帮助赢得第二次世界大战

重点(Top highlight)

定理 (The theorem)

惯常做法 (Frequentist approach)

贝叶斯定理_贝叶斯定理如何帮助赢得第二次世界大战相关推荐

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