Did Thomas Bayes truly develop Bayes' theorem in an effort to rebut David Hume's arguments against miracles?

Question

Perhaps this is a question better suited for Skeptics.SE, but it has a significant overlap with philosophy nonetheless. The context is this article:

Bayes’ theorem began as a defense of Christianity

Jordana Cepelewicz, Nautilus, December 20, 2016, on the Christian roots of Bayesian statistics:

Presbyterian reverend Thomas Bayes had no reason to suspect he’d make any lasting contribution to humankind. Born in England at the beginning of the 18th century, Bayes was a quiet and questioning man. … Yet an argument he wrote before his death in 1761 would shape the course of history. It would help Alan Turing decode the German Enigma cipher, the United States Navy locate Soviet subs, and statisticians determine the authorship of the Federalist Papers. Today it has helped unlock the secrets of the brain.

It all began in 1748, when the philosopher David Hume published An Enquiry Concerning Human Understanding, calling into question, among other things, the existence of miracles. According to Hume, the probability of people inaccurately claiming that they’d seen Jesus’ resurrection far outweighed the probability that the event had occurred in the first place. This did not sit well with the reverend.

Inspired to prove Hume wrong, Bayes tried to quantify the probability of an event. He came up with a simple fictional scenario to start: Consider a ball thrown onto a flat table behind your back. You can make a guess as to where it landed, but there’s no way to know for certain how accurate you were, at least not without looking. Then, he says, have a colleague throw another ball onto the table and tell you whether it landed to the right or left of the first ball. If it landed to the right, for example, the first ball is more likely to be on the left side of the table (such an assumption leaves more space to the ball’s right for the second ball to land). With each new ball your colleague throws, you can update your guess to better model the location of the original ball. In a similar fashion, Bayes thought, the various testimonials to Christ’s resurrection suggested the event couldn’t be discounted the way Hume asserted.

In 1767, Richard Price, Bayes’ friend, published “On the Importance of Christianity, its Evidences, and the Objections which have been made to it,” which used Bayes’ ideas to mount a challenge to Hume’s argument. “The basic probabilistic point” of Price’s article, says statistician and historian Stephen Stigler, “was that Hume underestimated the impact of there being a number of independent witnesses to a miracle, and that Bayes’ results showed how the multiplication of even fallible evidence could overwhelm the great improbability of an event and establish it as fact.”

Wikipedia suggests the same, but in a less conclusive manner:

In his later years he took a deep interest in probability. Historian Stephen Stigler thinks that Bayes became interested in the subject while reviewing a work written in 1755 by Thomas Simpson,[10] but George Alfred Barnard thinks he learned mathematics and probability from a book by Abraham de Moivre.[11] Others speculate he was motivated to rebut David Hume's argument against believing in miracles on the evidence of testimony in An Enquiry Concerning Human Understanding.[12] His work and findings on probability theory were passed in manuscript form to his friend Richard Price after his death.

Both articles reference this third article as their source, which in turn references the essay "On the Importance of Christianity, its Evidences, and the Objections which have been made to it" by Richard Price. With some googling I found more details on this essay: Richard Price, "On the Importance of Christianity, Its Evidences, and the Objections Which Have Been Made to It," dissertation 4, section 2, in Four Dissertations (London, 1767). A free version of Four Dissertations is available here. Dissertation 4, section 2, starts on page 384. I haven't had the time yet to review it in detail and find an explicit quote supporting the claim that Thomas Bayes defended miracles using Bayes' theorem. And I'm not aware of any other source that could potentially support this claim.

So my question is aimed at those who happen to be more knowledgeable in the topic. Did Thomas Bayes truly come up with his famous theorem in an effort to defend miracles from David Hume's objections?

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Note: I've asked a separate question about the validity of this argument: Can Bayes' theorem be used non-fallaciously to argue for miracles?

Answer 1

This is only a partial answer, but:

It's worth noting that Price's argument is not chiefly a Bayesian argument about the independence of testimony--that seems to be introduced by Stigler (or, more likely, I'm missing something very obvious--I'd appreciate if anybody can correct me):

The basic probabilistic point was that Hume underestimated the impact of there being a number of independent witnesses to a miracle, and that Bayes’s results showed how the multiplication of even fallible evidence could overwhelm great improbability of an event and establish it as fact. -- Stigler (2013)

Perhaps I'm missing something, but I don't think that's Price's basic point. Here's Earman 2002, which Stigler cites:

Hume makes some nods to the importance of multiple witnessing, but he seems not to have been aware of how powerful a consideration it can be. In fairness, the power was clearly and fully revealed only in the work of Laplace (1812, 1814) and more especially the work of Babbage (1838), whose Ninth Bridgewater Treatise devotes a chapter to a refutation of Hume's argument against miracles.

Note that Price (who is mentioned elsewhere in the paper) is absent. For the record, Babbage does clearly makes a Bayesian argument for multiple independent testimonies.

If independent witnesses can be found, who speak truth more frequently than falsehood, it is always possible to assign a number of independent witnesses, the improbability of the falsehood of whose concurring testimony shall be greater than that of the improbability of the miracle itself. -- Babbage 1838

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Returning to Price, here's the point where Price's footnote that cites the Bayes/Price letter. (The * is a footnote that cites Bayes to argue that a finite number of observations cannot give us certainty):

Upon observing, that any natural event has happened often or invariably, we have only reason to expect that it will happen again, with an assurance proportioned to the frequency of our observations. But, we have no absolute proof that it will happen again in any particular future trial; nor the least reason to believe that it will always happen*. For ought we know, there may be occasions on which it will fail... -- Price 1767

So it isn't about independent testimony, it's instead deployed as an argument against certainty. And the rest of the argument is that we regularly view human testimony as extremely convincing, even against long odds:

One action, or one conversation with a man, may convince us of his integrity and induce us to believe his testimony, though we had never, in a single instance, experienced his veracity. His manner of telling his story, its being corroborated by other testimony, and various particulars in the nature and circumstances of it, may satisfy us that it must be true. We feel in ourselves, that a regard to truth is one principle in human nature; and we know, that there must be such a principle in every reasonable being, and that there is a necessary repugnancy between the perception of moral distinctions and deliberate falsehood. To this, chiefly, is owing the credit we give to human testimony. And from hence, in particular, must be derived our belief of veracity in the Deity. -- p. 399

But let us take a higher case of this kind. The improbability of drawing a lottery in any particular assigned manner, independently of the evidence of testimony, or of our own senses, acquainting us that it has been drawn in that manner, is such as exceeds all conception. And yet the testimony of a newspaper, or of any common man, is sufficient to put us out of doubt about it. Suppose here a person was to reject the evidence offered him on the pretence, that the improbability of the falsehood of it is almost infinitely less than that of the event. -- p. 411