托马斯·贝叶斯Thomas Bayes

托马斯·贝叶斯的父亲Joshua Bayes是英格兰最早被任命的三位不从国教牧师之一。他于1694年被任命，并搬到距伦敦约25英里的博温顿的Box 桑德斯·麦克兰恩教堂。贝叶斯的母亲是Anne Carpenter。当贝叶斯年幼时，全家搬到了伦敦的萨瑟克，在那里Joshua成为圣贝叶斯教堂的助理，同时也是霍尔本Leather 麦克兰恩教堂的助理。贝叶斯是他父母七个孩子中的长子，四个男孩和三个女孩。

在[1]中，Hacking声称贝叶斯接受的是私人教育，他说这对于当时不从国教牧师的儿子来说似乎是必要的。如果是这样的话，那么关于他的导师就一无所知，但Barnard在[6]中指出了一个有趣的可能性：他可能由亚伯拉罕·棣莫弗辅导过，后者当时确实在伦敦提供私人辅导。然而其他历史学家（例如见[21]）认为贝叶斯接受了为牧师职业准备的自由教育。那样的话，他很可能就读于Tenter Alley的Fund Academy，那是贝叶斯住所附近唯一有正确宗教联系的学校。

我们确实知道，1719年贝叶斯进入爱丁堡大学学习逻辑学和神学。如果他想不出国就能接受教育，就必须选择一所苏格兰大学，因为当时不信奉国教者不被允许进入牛津或剑桥。爱丁堡大学保存至今的记录显示，他于1721年1月14日以“马太福音第7章第24-27节”为经文讲道，又于1722年1月20日以“马太福音第11章第29-30节”为经文讲道。他离开大学时是一名试用牧师，但此时尚未被正式任命。某个时候贝叶斯一定学过数学，但没有证据表明他是在爱丁堡大学学的。不过，他确实有机会在爱丁堡学习数学，而且他在34岁时写道：

我很久以前就认为，流数法的第一原理和规则需要更充分、更清晰的解释和证明……

这无疑表明他的兴趣可以追溯到学生时代，或者也许是此后不久。

贝叶斯被按立为牧师，像他父亲一样是一位不奉国教派的牧师，起初在霍尔本协助他的父亲。大约在1733年，在前任牧师弗瑞兹·约翰 Archer去世后，他成为位于伦敦东南35英里的坦布里奇韦尔斯的长老会教堂的牧师。然而，有证据表明他在1733年之前就与坦布里奇韦尔斯有联系。一份报告指出（例如参见[4]，其中引用了这一点）：-

约翰 Archer于1733年去世，但贝叶斯在1730年接替了他，贝叶斯是一个有相当成就的人。看来贝叶斯于1728年离开坦布里奇韦尔斯，并于1731年返回。在此期间，他在伦敦的Leather 麦克兰恩长老会教堂，他的父亲是该教堂的牧师。

1746年8月24日，威廉·惠斯顿描述与贝叶斯共进早餐，他说贝叶斯是：

……一位在坦布里奇韦尔斯的异议派牧师，也是汉弗莱·迪顿先生的继任者，虽然不是直接继任，并且像他一样是一位非常优秀的数学家。

贝叶斯 显然在1749年试图从牧师职务中退下来，但直到1752年才真正退休，期间一直担任坦布里奇韦尔斯的牧师，退休后仍继续住在坦布里奇韦尔斯。

贝叶斯 在1764年发表于Philosophical Transactions of the Royal Society of London的Essay towards solving a problem in the doctrine of chances中阐述了他的theory of probability。这篇论文是由贝叶斯的朋友理查德·普莱斯寄给皇家学会的，他写道：-

我现在寄给你一篇论文，这是在我已故朋友贝叶斯先生的文件中找到的，在我看来，这篇论文很有价值……在他为这篇论文所写的引言中，他说，他最初思考这个主题时的目的，是找出一种方法，使我们能够判断一个事件在给定情况下发生的概率，假设我们对它一无所知，只知道在相同情况下它已经发生了若干次，并且失败了另外若干次。

Dale在[4]中写道：-

读者可以从这篇论文中期待什么？关于概率，他当然会期待某种版本的所谓‘贝叶斯定理’：这种期待确实会得到满足。此外，他还会看到对二项分布的清晰讨论，如果他进一步深入探究，还会发现……涉及条件概率的概率逻辑结果的首次出现。这篇论文对数学家来说，在评估不完全贝塔函数方面应该是有意义的。我们还注意到，贝叶斯和理查德·普莱斯在这里以及补编中对各种积分的近似使用，以及对进行这种近似时所产生的误差问题的关注。因此，这篇论文主要地，也许公正地，因解决了贝叶斯提出的问题而被记住，但也应该因其对纯数学的贡献而被记住。

贝叶斯的结论在1781年的一篇回忆录中被皮埃尔·西蒙·拉普拉斯接受，被尼古拉·德·孔多塞重新发现（正如皮埃尔·西蒙·拉普拉斯提到的），并且一直未受质疑，直到乔治·布尔在Laws of Thought中提出疑问。从那时起，贝叶斯的方法就备受争议。

贝叶斯还写了一篇文章An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst（1736年），攻击乔治·伯克利对微积分逻辑基础的攻击。在序言中，贝叶斯给出了他撰写该文本的理由：-

我很久以前就认为，流数法的第一原理和规则需要比从最初那位无与伦比的作者或其任何追随者那里得到的更充分、更清晰的解释和证明；因此，当我发现该方法本身受到《分析家》那位才华横溢的作者如此激烈的反对时，我一点也不感到不快；如果他唯一的目的只是要把这一点提出来公平讨论，即通过流数法进行的证明是否真正科学，我会衷心赞扬他的行为，并认为他甚至值得数学家们本人的感谢。但是，他把这场辩论置于令人反感的境地，将其说成对宗教利益有影响，我认为这确实毫无道理，而且极不明智。

贝叶斯写道，乔治·伯克利：-

……将数学家之间的争论和争议说成是贬低他们方法的证据：并且……他将逻辑学和形而上学说成是能打开他们的眼睛，使他们摆脱困境的恰当工具。……如果任何科学的教授们的争论贬低了该科学本身，那么逻辑学和形而上学比数学受到更大的贬低，那么，为什么如果我半瞎，就必须让一个完全看不见的人做我的向导呢？

贝叶斯于1742年当选为皇家学会的会士，尽管当时他没有发表过任何数学著作，实际上在他有生之年也没有以他自己的名字发表过任何著作，上面提到的关于流数的文章是匿名发表的。另一篇关于渐近级数的数学出版物在他去世后出现，其中他表明斯特林和亚伯拉罕·棣莫弗给出的$\log z!$的级数无效，因为它是发散的。

还有几件其他的数学作品从贝叶斯流传到我们这里，我们现在来看其中一些。我们提到的第一件是他写的一封信，可能是在1756年左右。贝叶斯写道：-

你可能记得几天前我们谈到托马斯·辛普森先生的尝试，他试图表明在几次天文观测之间取平均值，而不是信赖一次精心进行的单一观测，以便减小由仪器和感觉器官的不完善所引起的误差，具有很大的优势。

事实上，托马斯·辛普森犯了与法国人近五十年后使用让-夏尔·德博尔达的复测圆时相同的错误，即相信通过进行多次观测，可以使观测中的误差小到人们所希望的程度。然而，贝叶斯意识到情况并非如此，并在他的信中写道：-

现在，由仪器和感觉器官的不完善所引起的误差，仅仅通过增加观测次数就应被减小到零或接近于零，这在我看来极其难以置信。相反，你使用不完善的仪器进行的观测越多，似乎你结论中的误差就越会与所使用仪器的不完善程度成正比……

在[4]中，一本几乎可以肯定是由贝叶斯写的笔记本被详细检查。这本笔记本包含相当多的数学工作，包括对概率、三角学、几何学、方程求解、级数和微分积分的讨论。还有关于自然哲学的部分，其中贝叶斯考察了包括电学、光学和天体力学在内的主题。

Thomas Bayes' father, Joshua Bayes, was one of the first six Nonconformist ministers to be ordained in England. He was ordained in 1694 and moved to Box Lane Chapel, Bovington, about 25 miles from London. Thomas's mother was Anne Carpenter. The family moved to Southwark, London, when Thomas was young and there Joshua became an assistant at St Thomas's and also an assistant at the Chapel in Leather Lane, Holborn. Thomas was the eldest of his parents seven children, four boys and three girls.

In [1] Hacking claims that Thomas was educated privately, something he says appears necessary for the son of a Nonconformist minister at that time. If this is the case, then nothing is known of his tutors but Barnard in [6] points out the intriguing possibility that he could have been tutored by de Moivre who was certainly giving private tuition in London at the time. However other historians (see for example [21]) suggest that Thomas received a liberal education for the ministry. In that case it is likely that he attended Fund Academy in Tenter Alley which was the only school with the right religious connections near where Bayes lived.

We do know that in 1719 Bayes matriculated at the University of Edinburgh where he studied logic and theology. He had to choose a Scottish university if he was to obtain his education without going overseas since, at this time, Nonconformists were not allowed to matriculate at Oxford or Cambridge. Records which still survive at the University of Edinburgh record that he gave the homily on 14 January 1721 with the text being "Matthew Chapter 7 verses 24-27", and again on 20 January 1722 with the text being "Matthew Chapter 11 verses 29-30". He left the University as a probationer, but he was not ordained at this stage. At some time Bayes must have studied mathematics but there is no evidence that he did so at Edinburgh University. However, he certainly had the opportunity to study mathematics at Edinburgh and when he wrote at age 34:-

I have long ago thought that the first principles and rules of the method of Fluxions stood in need of more full and distinct explanation and proof ...

he certainly suggests that his interest went back to his student days or perhaps shortly afterwards.

Thomas Bayes was ordained, a Nonconformist minister like his father, and at first assisted his father in Holborn. In about 1733 he became minister of the Presbyterian Chapel in Tunbridge Wells, 35 miles southeast of London, on the death of the previous minister John Archer. There is, however, evidence that he was associated with Tunbridge Wells before 1733. One report states (see for example [4] where this is quoted):-

John Archer died in 1733 but was succeeded by Thomas Bayes in 1730, who was a man of considerable attainment. It appears that Thomas Bayes left Tunbridge Wells in 1728 and returned in 1731. During this time he was at Leather Lane Presbyterian Church, London, where his father was pastor.

On 24 August 1746 William Whiston describes having breakfast with Bayes who he says is:-

... a dissenting Minister at Tunbridge Wells, and a Successor, though not immediate, to Mr Humphrey Ditton, and like him a very good mathematician.

Bayes apparently tried to retire from the ministry in 1749 but remained minister at Tunbridge Wells until 1752 when he did retire, but continued to live in Tunbridge Wells.

Bayes set out his theory of probability in Essay towards solving a problem in the doctrine of chances published in the Philosophical Transactions of the Royal Society of London in 1764. The paper was sent to the Royal Society by Richard Price, a friend of Bayes', who wrote:-

I now send you an essay which I have found among the papers of our deceased friend Mr Bayes, and which, in my opinion, has great merit... In an introduction which he has writ to this Essay, he says, that his design at first in thinking on the subject of it was, to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times.

Dale writes in [4]:-

What may the reader expect to find in this Essay? As regards probability, he will expect, of course, some or other version of what has become known as 'Bayes's theorem': and such expectation will indeed be met. In addition he will find a clear discussion of the binomial distribution and if he should probe even deeper he will find ... the first occurrence of a probability logic result involving conditional probability. The Essay should be of interest to mathematicians for the evaluation of the incomplete beta-function. We note too the use of approximations to various integrals made here and in the Supplement by both Bayes and Price, and the attention paid to the question of the error incurred in the making of such approximation. The Essay, then, mainly, and perhaps justly, remembered for the solution of the problem posed by Bayes, should also be remembered for its contribution to pure mathematics.

Bayes's conclusions were accepted by Laplace in a 1781 memoir, rediscovered by Condorcet (as Laplace mentions), and remained unchallenged until Boole questioned them in the Laws of Thought . Since then Bayes' techniques have been subject to controversy.

Bayes also wrote an article An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst (1736) attacking Berkeley for his attack on the logical foundations of the calculus. In the Preface Bayes gives his reasons for writing the text:-

I have long ago thought that the first principles and rules of the method of Fluxions stood in need of more full and distinct explanation and proof, than what they had received either from their first incomparable author, or any of his followers; and therefore was not at all displeased to find the method itself opposed with so much warmth by the ingenious author of the Analyst; and had it been his only design to bring this point to a fair issue, whether a demonstration by the method of Fluxions be truly scientific or not, I should have heartily applauded his conduct, and have thought he deserved the thanks even of the Mathematicians themselves. But the invidious light in which he has put this debate, by representing it as of consequence to the interests of religion, is, I think, truly unjustifiable, as well as highly imprudent.

Bayes writes that Berkeley:-

...represents the disputes and controversies among mathematicians as disparaging the evidence of their methods: and ... he represents Logics and Metaphysics as proper to open their eyes, and extricate them from their difficulties. ... If the disputes of the professors of any science disparage the science itself, Logics and Metaphysics are much more disparaged than Mathematics, why, therefore, if I am half blind, must I take for my guide one that can't see at all?

Bayes was elected a Fellow of the Royal Society in 1742 despite the fact that at that time he had no published works on mathematics, indeed none were published in his lifetime under his own name, the article on fluxions referred to above was published anonymously. Another mathematical publication on asymptotic series appeared after his death where he showed that the series for $\log z!$ given by Stirling and de Moivre, was not valid since it diverged.

There are a few other pieces of mathematics which have come down to us from Bayes and we now look at some of these. The first we mention is a letter which he wrote, probably around 1756. Bayes wrote:-

You may remember a few days ago we were speaking of Mr Simpson's attempt to show the great advantage of taking the mean between several astronomical observations rather than trusting to a single observation carefully made, in order to diminish the errors arising from the imperfection of instrument and the organs of sense.

In fact Simpson had made the same error that the French would make nearly fifty years later with Borda's repeating circle, in believing that one could make the error in the observation as small as one desired by making multiple observations. However, Bayes realised that this was not so and wrote in his letter:-

Now that the errors arising from the imperfection of the instrument and the organs of sense should be thus reduced to nothing or next to nothing only by multiplying the number of observations seems to me extremely incredible. On the contrary the more observations you make with an imperfect instrument the more it seems to be that the error in your conclusion will be proportional to the imperfection of the instrument made use of ...

In [4] a notebook which was almost certainly written by Bayes is examined in detail. This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus. There are also sections on natural philosophy in which Bayes looks at topics which include electricity, optics and celestial mechanics.