In my previous post (25. The Sestercentennial of Thomas Bayes and Richard Price), I had introduced Stephen Stigler’s argument that Thomas Bayes’ 1763 paper is connected to David Hume. In particular, Stigler has suggested that Bayes’ work, as disseminated by Price, represented a mathematical challenge to Hume’s skeptical observations on matters of religion.

Stigler begins his paper with the interesting finding that Bayes’ 1763 essay may have had two titles. The accepted title of Bayes’ essay, as published in *Philosophical Transactions*, was *An Essay towards Solving a Problem in the Doctrine of Chances*. The other title of Bayes’ work, however, may have been *A Method of Calculating the Exact Probability of All Conclusions founded on Induction*. This was either the original title of Bayes’ essay prior to publication or Price chose to rename Bayes’ essay following publication.

Evidence for this alternate title comes from an offprint that followed the publication of Bayes’ essay. (In more current terms, an offprint is akin to a journal reprint but under its own cover.). This alternative title is also described in a footnote in Price’s *Four Dissertations* (1767) as well as in Volume 7 of Rees’ *New Cyclopaedia* (1807).

For Stigler, this alternate title is meaningful and he suggests that the intent of Bayes was to address the issue of induction. Following this assumption, Stigler then constructs the following possible sequence of events:

[One]. In 1748, Hume publishes his essay *Of Miracles* in his *Philosophical Essays Concerning Human Understanding*. Hume’s essay that challenged witness testimony to miracles would have been controversial among Christian clergy and members of the church.

[Two]. Bayes attempts to address Hume’s argument in 1748 or 1749 by working on a method of applying probability to unknown causes (or induction). Based on extant notes, Stigler argues that Bayes would have most likely completed his calculations prior to December 31, 1749. Stigler then presumes that Bayes put this work aside due to it being either insufficient or not satisfactory to Bayes.

[Three]. Sometime between 1749 and just prior to his death, Bayes discusses the above work with Richard Price. Stigler emphasizes Bayes’ and Price’s shared religious beliefs as evidence for a likely acquaintance between Bayes and Price as well as Bayes bequeathing £100 to Price in Bayes’ will. Stigler notes that Bayes may have also described his method to David Hartley in 1749. If Bayes shared his work with Hartley, Stigler argues, then it would make sense that Bayes would have also shared his work with his more closer acquaintance, Richard Price.

[Four]. Bayes dies on April 7, 1761. With his prior knowledge of Bayes’ work, Price obtains Bayes’ manuscript from Bayes’ private documents. Price then spends the next two years expanding and editing Bayes’ manuscript with the explicit purpose of challenging Hume.

[Five]. Price delivers Bayes work to the Royal Society on December 23, 1763 and further extends this work with a second paper read to the Society one year later.

[Six]. According to Stigler (and colorfully), in 1767, the final weapon is deployed — Price’s work *Four Dissertations*. In his fourth dissertation, Price argues against Hume and attempts to show mathematically how even improbable events (miracles) can be established through accumulation of evidence from independent witnesses. Price relies on Bayes’ essay as his source of authority.

And the circle is now complete from Hume to Bayes to Price to Hume.

Stigler acknowledges that it is speculative whether Bayes’ original intention in carrying out his calculations was based on a direct response to Hume. He accepts that evidence to support this argument is scant given that Hume is not referred to in Bayes’ work and that few documents of Bayes have survived. However, Stigler argues that the timing of Hume’s work and the timing of Bayes’ initial calculations on probability are compelling.

Indirect evidence that Bayes may have completed his calculations prior to 1749 comes from information contained in Hartley’s work *Observations on Man*. Here, Hartley refers to an “ingenious friend” who had demonstrated a method of calculating the probability of an unknown cause given only the observation of its effects. A number of statistical historians (although not all) have suggested that the ingenious friend mentioned by Hartley was, in fact, Bayes. If so, suggests Stigler, then the bulk of Bayes’ work would have been completed following the publication of Hume’s essays in 1748 and prior to Hartley’s 1749 publication.

With all due respect to a Stigler, I am having difficulty accepting his argument in its entirety. In the next post, I will begin to review his thesis item by item. We will begin at the beginning and first discuss induction.

- [
*Note.*If you are a style junkie you may have noticed that when I use the possessive form of Bayes I use Bayes’. In his article, Stigler uses Bayes’s. You will get different opinions on which form is correct, particularly when dealing with proper nouns. Overall, the trend seems to be moving toward Stigler’s use as the preferred method. I personally find that style awkward but I accept that things change. For now, however, I will stay with the older method.]

- Stigler, S. M. (2013). The true title of Bayes’s essay.
*Statistical Science*,*28*, 283-288. - Chances, doctrine of. (1807). In
*The cyclopeadia, or universal dictionary of arts, sciences, and literature (Vol. 7, para. 2)*. London: Longman, Hurst, Rees, Orme, & Brown.