Thomas Bayes: A Life in Five Chapters

Chapter 1 · Shut Out

A Dissenter's education

c. 1701 – 1734 · London · Edinburgh · Tunbridge Wells

Thomas Bayes was born around 1701 — the year is not certain — the eldest of seven children of Joshua Bayes, one of the first Nonconformist ministers publicly ordained in England after the Toleration Act of 1689.

Being a Dissenter had a specific consequence. Oxford and Cambridge required subscription to the Articles of the Church of England, so they were closed to him. He went instead to the University of Edinburgh, which was not, and studied logic and theology there from 1719.

This is not a footnote. A whole class of educated Englishmen were excluded from the two English universities for a century and a half, and the Dissenting academies and the Scottish universities that took them produced a disproportionate share of the period's scientific and industrial talent. Bayes is one instance of a large pattern.

He was ordained, assisted his father, and around 1734 became minister of the Presbyterian meeting-house at Tunbridge Wells in Kent, where he stayed for most of his working life. Tunbridge Wells was a spa town: fashionable, full of visitors taking the waters, and — relevantly — full of gambling.

He never married. He published two works in his lifetime and neither carried his name. And that is very nearly the whole of what is known about his life. There are no letters of consequence, no diary, no anecdotes, and no authenticated portrait; the picture reproduced everywhere is of an unidentified clergyman, attached to his name in a 1936 book about insurance on no stated authority, and he died in 1761, which makes a nineteenth-century sitter impossible.

Why this matters

The exclusion of Dissenters from the English universities shaped who did English science, and the near-total absence of biographical record is itself a fact about the period.

A life almost entirely without documents. What would you ask him?

Ask Bayes

• “What did it mean to be shut out of Oxford and Cambridge?”
• “Why Edinburgh?”
• “What was a minister's life at Tunbridge Wells like?”
• “Did the gambling at the spa have anything to do with it?”
• “Does it trouble you that no likeness of you survives?”

Chapter 2 · The Ghosts of Departed Quantities

Answering Bishop Berkeley, 1736

1731 – 1742 · Tunbridge Wells · London

In 1734 George Berkeley, bishop and philosopher, published The Analyst, subtitled a discourse addressed to an infidel mathematician. It is an attack on the foundations of Newton's calculus, and it is devastating.

Berkeley's point was that the method works by taking a quantity, dividing by it, and then setting it to nothing — which is either legitimate or not, and cannot be both. If the quantity is something, you may not discard it; if it is nothing, you may not have divided by it. He called the vanishing increments the ghosts of departed quantities and asked why mathematicians who demanded rigour of theologians accepted this from themselves.

The attack landed. It was not answered adequately for a century, until the nineteenth century built a proper theory of limits.

In 1736 Bayes published, anonymously, An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians against the Objections of the Author of the Analyst. It is careful and it concedes a good deal: Bayes accepts that the standard presentations are not rigorous and argues that the method can be given a sound footing, working through the reasoning to show what a defensible version would look like.

It is probably the reason he was elected a Fellow of the Royal Society in 1742, at a point when he had published no mathematics under his own name at all. The nomination described him as a gentleman well skilled in geometry and all parts of mathematical and philosophical learning.

The habit of mind on display — take the objection seriously, concede what is true in it, and reconstruct the foundation — is exactly the habit that produced the essay he did not publish.

Why this matters

The best-known attack on the calculus was answered by an anonymous country minister, and how somebody handles an objection tells you what kind of thinker they are.

A bishop's attack, and a minister's answer. What would you ask him?

Ask Bayes

• “What was Berkeley's objection to the calculus?”
• “Was he right?”
• “How did you answer him?”
• “Why publish anonymously?”
• “How did the Royal Society come to elect you?”

Chapter 3 · The Table and the Ball

The question nobody had asked

c. 1750s · Tunbridge Wells

By the middle of the eighteenth century the mathematics of chance was well developed in one direction. Given a known chance — a fair die, a bent coin whose bias you have been told — de Moivre's Doctrine of Chances could tell you how a run of throws was likely to come out.

Bayes asked the other question, and it is the one an actual human being faces. You do not know the chance. You have the throws. What may be said about the chance?

His approach is characteristic: rather than argue in generalities, he builds an arrangement you can picture. A level table. A ball is rolled across it and comes to rest somewhere; where it stops divides the table into a left part and a right part, and the proportion of the table lying to its left is a chance — unknown to you, because you did not see. Now more balls are rolled, and for each you are told only whether it came to rest to the left or the right of the first.

From that record he reasons backwards to where the first ball must be. If ten balls follow and eight land to the left, the first ball is probably well over to the right; how probably, and within what range, is what he works out.

The reasoning has one deep difficulty and Bayes saw it. To get an answer you must say something about where the first ball was likely to be BEFORE any of the others were thrown. Bayes takes it to be equally likely to have stopped anywhere, which is defensible for a ball rolled across a level table and much less obviously defensible for anything else. What to assume before the evidence arrives is precisely the problem that made his method controversial for two hundred years.

He wrote it up. He did not publish it, and there is no evidence he showed it to anybody.

Why this matters

The question of what a cause may be, given the evidence, is the one science actually asks — and reasoning from a picture rather than a formula is how it was first answered.

A table, a ball, and a question turned round. What would you ask him?

Ask Bayes

• “What question were you asking that nobody had?”
• “How does a ball on a table settle a question about chance?”
• “What has to be assumed before any evidence arrives?”
• “Is it safe to assume the ball was equally likely to stop anywhere?”
• “Why reason from a picture rather than from a formula?”

Chapter 4 · Richard Price Finds It

1763, two years too late

1761 – 1763 · Tunbridge Wells · London

Bayes died on 7 April 1761 and was buried at Bunhill Fields, the Nonconformist burial ground in London. In his will he left a hundred pounds to be divided between John Boyl and Richard Price — Price being a fellow Dissenting minister, and later a well-known moral philosopher and political radical, a friend of Franklin and a defender of the American Revolution.

Price was asked by the family to go through the mathematical papers. He found the essay, understood what it was, and did something scholars are not always willing to do for a dead friend: he worked on it. He wrote a preface, added an appendix applying the reasoning, and on 23 December 1763 read it to the Royal Society. It was published in the Philosophical Transactions the following year.

Price's preface makes a claim that Bayes had not: that the method bears on the foundations of inductive reasoning generally — on the question of why, having seen the sun rise a great many times, one is entitled to expect it tomorrow. That is a much bigger claim than anything in the essay itself, and it is Price's, not Bayes's.

The paper made no immediate impression. Ten years later Laplace, who almost certainly had not read it, developed the same inverse reasoning independently, in a far more general form, and did enormous amounts with it; for most of the nineteenth century what we now call the Bayesian approach was called the method of inverse probability and attributed to Laplace.

Bayes's name became attached to it later, and now it is everywhere.

Why this matters

Work that reaches the world at all often depends on somebody else caring enough to carry it there — and on what they add on the way.

A friend going through the papers of a dead man. What would you ask him?

Ask Bayes

• “Why did you leave the essay unpublished?”
• “Who was Richard Price?”
• “Price claimed the essay bore on induction. Did you claim that?”
• “How much of the famous result is actually Laplace's?”
• “Would you have wanted it printed?”

Chapter 5 · Two Centuries of Argument

Derided, then indispensable

1763 – now · London · Bletchley Park · everywhere

What happened to the essay's method after Bayes is one of the strangest careers in the history of ideas.

Laplace used it heavily and productively. Then, through the later nineteenth and much of the twentieth century, it was attacked and largely abandoned by professional statisticians. The objection was always the same one Bayes had seen: the method requires you to say what you believed before the evidence arrived, and that looks like importing an opinion into a calculation that is supposed to be objective. Fisher and others built an entire alternative statistics designed to avoid it, and to use the word Bayesian was for decades to mark yourself as unsound.

Meanwhile it kept working. Turing and Good used the reasoning at Bletchley Park to weigh evidence about Enigma settings, in secret, with results that could not be published for thirty years. Insurance actuaries used it. It was used to locate a hydrogen bomb lost off the Spanish coast in 1966 and a submarine lost in 1968.

From the 1980s, cheap computation made the calculations feasible at scale, and the position reversed. It is now central to statistics, to machine learning, to medical diagnosis and to a great deal of science. Every spam filter uses it. So does any doctor who correctly interprets a positive test result for a rare disease — the classic case where the answer is wildly counter-intuitive unless you take the shrinking of the world seriously.

The argument about what to assume before the evidence arrives is not settled. It is simply that everybody now agrees it has to be faced rather than avoided.

A country minister with no portrait, no published mathematics under his own name, and no evidence that he told a single person, wrote the paper that changed how the world reasons from evidence to cause.

Why this matters

An idea can be right, ignored, derided and then indispensable, and the objection that got it derided was the one its author had already identified.

Two hundred years out of favour, and now everywhere. What would you ask him?

Ask Bayes

• “Why was your method attacked for so long?”
• “Is it a weakness to have to say what you believed beforehand?”
• “What would you make of it being used to break Enigma?”
• “Why do doctors get positive test results wrong?”
• “What do you make of your name being on all of it?”

What Bayes changed

The reasoning in Bayes's unpublished essay — from observed outcomes back to the chance that produced them — is now how a great deal of science, medicine and machine learning updates a belief in the light of evidence. It was used at Bletchley Park against Enigma, to find a lost hydrogen bomb in 1966, and it underlies every spam filter and every properly interpreted diagnostic test. His published defence of Newton's calculus against Berkeley was a serious answer to a serious attack, and probably earned him his Fellowship of the Royal Society.

A debate that continues

He did not publish the essay and there is no evidence he showed it to anybody; Richard Price found it after his death, added a preface claiming more for it than Bayes had, and read it in 1763. Laplace developed the same reasoning independently a decade later and did far more with it, so how much of the modern method is Bayes's is genuinely disputed. The difficulty he identified — what may be assumed before any evidence arrives — kept the approach out of favour for most of two centuries and is still argued about. No authenticated portrait of him exists; the image reproduced everywhere is of an unidentified clergyman.

Keep exploring — ask Bayes

• “What may honestly be assumed before any evidence has arrived?”
• “Is reasoning from evidence back to cause a different kind of thinking from reasoning forwards?”
• “How much does a discovery depend on somebody else deciding it matters?”

Related lives

• Blaise Pascal — Who Carried A Barometer Up A Mountain
• Alan Turing — Father of Computer Science
• Jacob Bernoulli — Against My Father's Will

Related themes

Conditional probability · Evidence and inference · The foundations of the calculus