Pioneers of Probability · Episode 11

This video is for informational and educational purposes only and does not constitute a solicitation or recommendation to buy or sell any security. The historical and mathematical concepts discussed are intended to illustrate the development of probability theory and its relevance to investing. Past performance is not indicative of future results. All investing involves risk, including the possible loss of principal. The examples provided are hypothetical and based on historical index data, not an actual investment. Index returns do not reflect the performance of any actual portfolio or the deduction of advisory fees. Content is AI assisted.

 

◆ The Question

No verified portrait of Thomas Bayes exists. The image most commonly attached to his name — a man in clerical collar and full wig, reproduced in a 1936 publication and reprinted thousands of times since — has been questioned by historians who note that the collar style and wig do not match mid-eighteenth century nonconformist fashion. We may have been looking at the wrong face for nearly a century. It is a fitting irony for a man whose entire contribution to mathematics was about looking more carefully at the evidence.

What we do know about Thomas Bayes is this: he was born around 1701, the eldest son of Joshua Bayes, a Presbyterian minister in London. He studied logic and theology at the University of Edinburgh — one of the few universities open to nonconformists, who were barred from Oxford and Cambridge by the religious tests of the era. He assisted his father in his London congregation, then in 1731 became minister of the Presbyterian chapel at Mount Sion in Tunbridge Wells, a fashionable spa town in Kent, thirty-five miles from London. He remained minister there for roughly thirty years, until illness forced his retirement, and he died in Tunbridge Wells on April 7, 1761.

He was elected a Fellow of the Royal Society in 1742 — an honor typically requiring significant published scientific work — which tells us that his mathematical abilities were recognized by his peers even in his lifetime. But he published almost nothing under his own name. Two pamphlets survive: a defense of Newton's calculus, and a theological tract. The work that would make him permanently famous was never submitted for publication. It was found in his papers after his death and very nearly did not survive at all.

 

◆ The Insight

The mathematical problem Bayes was trying to solve had been circulating since Leibniz first framed it: how should a rational person revise their beliefs when new evidence arrives? The question sounds philosophical, but Bayes attacked it as mathematics. Given that an event has occurred, what is the probability that it was produced by a particular cause?

This is the inverse problem of probability — the problem of reasoning backwards from observed effects to probable causes. It is the problem a physician faces when interpreting a test result: the test is positive, but what is the probability the patient actually has the disease? It is the problem a judge faces evaluating testimony: the witness claims to have seen the defendant, but given everything else in the record, what is the probability the identification is correct? It is the problem an investor faces evaluating a track record: the manager has outperformed for three years, but given how many managers are trying, what is the probability this represents genuine skill rather than chance?

Classical probability, as Cardano, Pascal, and Bernoulli had developed it, moves forward: from known probabilities to predicted outcomes. Bayes wanted to move backward: from observed outcomes to updated beliefs about underlying probabilities. It was a genuinely different kind of question, and it required a genuinely different kind of answer.

What he discovered is that the answer involves combining three things: what you believed before you saw the evidence (the prior), how probable the evidence would be if your hypothesis were true (the likelihood), and how probable the evidence is overall across all hypotheses (the normalizing constant). The result is your updated belief after incorporating the evidence — the posterior.

 

◆ The Proof

The theorem on the reverse of the IFA MarketCoin® for Bayes states the relationship precisely: P(A|B) = P(B|A) × P(A) / P(B). The coin shows a Venn diagram with two intersecting circles labeled A and B, their overlap representing the joint probability, beneath the inscriptions "Belief Adjusts as Evidence Accumulates" and "Bayes' Theorem." In plain language: the probability of A given that B has occurred equals the probability of B given A, multiplied by your prior probability of A, divided by the overall probability of B.

Bayes illustrated his theorem with a thought experiment involving a billiard table: imagine rolling a ball onto the table and not looking where it stops. Then roll more balls, one at a time, and observe whether each one stops to the left or right of the first ball. Based only on these observations — left or right — how certain can you become about where the first ball stopped? Bayes showed that as the number of observations grows, your estimate converges toward the true position. Belief updating through evidence accumulation.

The manuscript describing this work was found after Bayes' death by his friend and fellow minister Richard Price, who had been left £100 in Bayes' will — a detail that speaks to the depth of their friendship. Price recognized immediately that he was holding something important. He spent two years editing the manuscript, writing an introduction explaining its significance, and corresponding with the mathematician John Canton to refine the presentation. On December 23, 1763, Price presented the work to the Royal Society. It was published in 1764 in the Philosophical Transactions under the title "An Essay Towards Solving a Problem in the Doctrine of Chances."

Without Price, the theorem disappears. It is one of the great acts of intellectual rescue in the history of science — a friend who understood what he was looking at, took the time to understand it fully, and made sure the world heard about it. The £100 bequest was repaid several thousand times over in the currency of enduring ideas.

 

◆ The Legacy

The theorem aroused little immediate interest. It was Pierre-Simon Laplace who independently rediscovered and fully formalized the Bayesian framework in the 1770s and 1780s, extending it from Bayes' specific geometric formulation into the general analytical tool we use today. The approach was sometimes called Bayesian-Laplacian inference in the nineteenth century, and then simply Bayesian inference in the twentieth.

The frequentist-Bayesian debate — between those who treat probability as a frequency of repeated events and those who treat it as a rational degree of belief — dominated statistics throughout most of the twentieth century, with the frequentist tradition dominant in academic statistics and the Bayesian approach gaining ground slowly through applications in computing, machine learning, and decision theory. Today, Bayesian inference powers spam filters, medical diagnostic algorithms, search and rescue systems, natural language models, and the updating mechanisms behind most modern artificial intelligence. Every time a system receives new data and revises its estimates accordingly, it is doing what Bayes described in a manuscript that spent two years in a dead man's desk drawer before anyone knew it existed.

 

◆ Your Money

Every investment decision begins with a prior — a belief held before the evidence is examined. You believe a fund manager has skill. You believe a particular asset class will outperform over the next decade. You believe the market is overvalued. These are priors, held with varying degrees of confidence.

Bayes' theorem tells you precisely how to update them. When a manager outperforms for one year, the appropriate update depends on two things: the strength of the evidence (how much one year of outperformance actually tells you about underlying skill) and the base rate (how often managers genuinely have persistent skill that persists). The base rate is low. One year of strong returns is weak evidence for genuine skill. Bayes' theorem says: update your prior, but not by much.

Most investors do the opposite. They see strong recent performance and dramatically revise their beliefs upward — pouring money into last year's winners, firing managers after a bad quarter, treating recent data as far more informative than it is. This is a systematic failure of Bayesian reasoning: overweighting the new evidence and ignoring the prior.

The coin's rim says: belief adjusts as evidence accumulates. That is the formula for rational investing, stated in five words. Not belief stays fixed despite evidence. Not belief swings wildly with every data point. Belief adjusts — proportionally, systematically, in the exact ratio that Bayes derived in a quiet study in Tunbridge Wells, in a manuscript that almost never made it out of the room.

 

Sources: Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society, 53, 370–418. McGrayne, S. B. (2011). The theory that would not die. Yale University Press. MacTutor History of Mathematics, University of St Andrews.

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