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How Bayes’ Math Rule Can Counter Unreasonable Skepticism

Mathematics is much more interesting if we know a bit about the players and their positions

Yesterday, we discussed the importance of Bayes’ rule in statistical reasoning. We used the example of a person who goes for a battery of screening tests and comes up positive for HIV. Let’s say she is surprised (and alarmed) because she is not at any known risk for HIV.

But, it turns out, the risk of false positives for the test is several times greater than the incidence of HIV in the population. In that case, it is reasonable for her to suspect—on a statistics science basis, not just wishful thinking—that the test is a false positive. The formula we used is part of Bayesian reasoning, originally developed by an eighteen-century British clergyman and mathematician Thomas Bayes (1702–1761), but now widely used to assess probabilities.

When Bayes first proposed his approach, it was not just another a dry theorem. It was a challenge to a powerful philosophy advocating extreme skepticism—the philosophy of David Hume (1711–1776) (pictured).

Bayes was not formally trained as a mathematician; he was a highly skilled amateur. He had written a book describing and defending the foundations of calculus when he was in his thirties. Formally, he studied logic and theology at the University of Edinburgh.

He was later ordained, and eventually became the minister at Presbyterian Chapel in Tunbridge Wells in 1733, where he remained until he retired 1752. He fell ill in 1755 and died in 1761. His work on probability was only publicized after his death by Richard Price (1723–1791), a friend and fellow clergyman who was also a pioneer insurance statistician.

So why did Bayes choose to clash with the famous philosopher Hume? Mathematics historian Stephen Stigler has been making the case that there was an underlying issue that is still relevant today.

In his iconic chapter “On Miracles” (1748) in the Enquiries concerning Human Understanding, Hume tried to use probability theory to show that reports of miracles change nothing regarding our understanding of human existence. This essay was widely distributed and was, no doubt, of concern to Christian clergy.

Based on his surviving notes, it appears that Bayes began his work on probability shortly after Hume’s essay was published, David Hartley (1705–1757) apparently referred to them in his 1749 book, Observations on Man, as Hartley’s “ingenious friend.”

Later, he met and became good friends with Richard Price. We don’t know what they discussed but we do know that Price sought out Bayes’ manuscript after his death and spent two years enlarging it and preparing it for publication. As Stigler points out, the manuscript concerned problems of induction (determining facts from evidence, as in our HIV example). And induction is precisely the problem that David Hume was addressing.

Price went on to publish a rebuke of Hume in his own book, Four Dissertations. Price shows, citing Bayes’ paper as an authority, that even improbable events such as miracles can be established by eyewitness testimony from multiple witnesses.

We cannot be sure that Bayes developed the theorems he did for the express purpose of countering Hume because he left no written record of that intention. It is most likely, however, that Bayes’ theorem and its implications with respect to Hume’s assertions formed part of his and Price’s discussions. So, of course, after Bayes’s death, Price went to considerable trouble to dig up and prepare his late friend’s manuscript to use in that philosophical fight, believing that his friend would want him to do so.

Many people see mathematics as a dry manipulation of numbers. It is often a philosophical duel played out with symbols instead of swords. And it is much more interesting if we know a bit about the players and their positions.

Note: Some readers may be surprised by the prominent role clergy played in pioneering math and science studies during the eighteenth and nineteenth centuries. Few commoners in those days could afford much education but clergy were provided with education. Many of them used the opportunity not only to perform their duties but to advance various disciplines, as Bayes did with statistics and Price did with insurance calculations (actuarial tables).

Here’s Part 1: Can an 18th century statistician help us think more clearly? Distinguishing between types of probability can help us worry less and do more. Bayes’ Rule can help us distinguish between types of probability if, for example, we have a positive screening test for a disease but no symptoms.

Jonathan Bartlett

Senior Fellow, Walter Bradley Center for Natural & Artificial Intelligence
Jonathan Bartlett is a senior software R&D engineer at Specialized Bicycle Components, where he focuses on solving problems that span multiple software teams. Previously he was a senior developer at ITX, where he developed applications for companies across the US. He also offers his time as the Director of The Blyth Institute, focusing on the interplay between mathematics, philosophy, engineering, and science. Jonathan is the author of several textbooks and edited volumes which have been used by universities as diverse as Princeton and DeVry.

How Bayes’ Math Rule Can Counter Unreasonable Skepticism