23. The Bayesian Belief: 1. Thomas Bayes and Richard Price

Over the past few months we have been wandering through the dense undergrowth of the man forest and have spent considerable time pondering testosterone.  We have been poking at the Androgen Deficiency in Aging Males (ADAM) questionnaire for no reason other than it serves as a useful prop to discuss test statistics and potential biases that may be present during the diagnostic process.  In doing so, we have focused on Bayesian statistics.

Often statistics are presented as ahistorical formulas that are seemingly dropped from the heavens.  Yet all statistics were created at a certain place and time and all were created for a specific reason. To best understand a statistic, it is important to understand its context and the purpose for which it was originally created.

The story of Bayesian statistics begins, not surprisingly, with a man named Bayes.

Thomas Bayes (1702-1761) was an English Presbyterian minister and Fellow of the Royal Society of London.  Bayes was not a prolific author and, in his lifetime, it has been presumed that he published just two pieces of work:  a theological essay on divine benevolence and an exploration of the work of Isaac Newton.  Both of these texts were published anonymously.  However, evidence that these works were written by Bayes stems from references made by Bayes’ contemporaries attesting to Bayes’ authorship and the tendency of readers to write Bayes’ name on the front of these publications.

Bayes Book

The last decade of Bayes’ life was marred by illness and Bayes died in 1761 at the age of 59.  In his will, Bayes bequeathed £100 (approximately £13000 in current purchasing power) to Richard Price.  Two years later, in 1763, Price submitted to the Royal Society work by Bayes that Price had presumably uncovered when reviewing Bayes’ unpublished documents.  Price titled that work An essay towards solving a problem in the doctrine of chances and this essay was published in the Society’s journal, Philosophical Transactions.  The full text of that publication is freely available on the Society’s website at http://rstl.royalsocietypublishing.org/content/53/370.full.pdf.  However, I warn you, it is not an easy read and it is, as so eloquently described by Stephen Fienberg, “remarkably opaque in notations and detail.”

Price and Bayes’ paper had little impact immediately following its publication.  It did, however, lead to Price being elected a member of the Royal Society in 1765 and marked the start of Price’s long and fruitful career in probability and statistics, actuarial science, theology, and political and religious activism.

Bayes’ end was Price’s beginning.

How exactly Bayes and Price were connected remains unclear.  Although Price was Bayes’ junior by 20 years, both men were Protestant nonconformists.  Nonconformists refused to accept the Anglican Book of Common Prayer and, as a consequence, were barred from attending university or holding public office.  In order to provide higher education to young nonconformist men, nonconformist ministers created dissenting academies intended to provide the equivalent of university training. One such academy was Moorfields whose principal from 1734 to 1744 was John Eames.  In 1742, Eames was a nominator for Bayes’ election to the Royal Society.  At the same time, Eames was also instructing a young man named Richard Price in mathematics among other subjects.  A number of authors have argued that it may have been through Eames that Bayes and Price became familiar with each other and that Price may have been an admirer of Bayes’ mathematical work.

Yet, evidence to support a academic or peer relationship between Bayes and Price is scant.  Neither Bayes or Price make personal reference to each other in their private documentation or diaries.  Similarly, while it is true that Bayes left £100 to Price, it is perhaps more accurate to say that Bayes left £200 in his will to be split evenly between Price and John Hoyle.  Nor did Bayes’ will stipulate on how that money should be spent.

Bayes Will

Instead, around the time Bayes’ was writing his will, Richard Price had recently succeeded John Hoyle as a minister at Newington Green — an important nonconformist place of worship. The Bayes’ family had a tradition of bequeathing money to ministers they admired.

Bayes Price Hoyle Newington Green

Using a contemporary perspective, we connect Bayes and Price through their joint work on probability.  This is the connection that has survived across 250 years.  However, if we place ourselves within those times that existed during Bayes’ and Price’s lives, it is religion that dominates and not mathematical oddities such as probability.  It was a time of Huguenot persecution and diaspora in France.  It was a time of struggle between Anglican, Catholic, and Protestant belief in England with England still coping with the aftermath and scars of civil war and regicide. What you believed in defined your life and, sometimes, your death.  Therefore, it is more probable that it was Bayes who posthumously honored Price and the connection between these two men was mostly likely one of shared spiritual belief.

If so, how is that Price came into possession of Bayes’ work?

The one reference that addresses this comes from William Morgan, Richard Price’s nephew.  In Morgan’s 1815 biography of his uncle’s life and achievements, Morgan indicates that Price was asked by Bayes’ family to review Bayes’ papers and unpublished writings.  According to Morgan, Price did much more than merely submit Bayes’ work and, instead, his uncle spent considerable time collating, editing, and clarifying Bayes’ papers.

Of most interest to Price was Bayes’ work on probability. To illustrate, in the 1763 volume of Philosophical Transactions there was not one, but two, posthumous articles by Bayes.  The first concerned Bayes’ work on infinite series and the second was his more famous work on probability.  If Price submitted Bayes’ letter regarding infinite series, apparently he did so with marginal edits.  In Bayes essay on probability, however, Price added a lengthy introduction and appendix.  Similarly, in the 1764 volume an additional article appears by Price that further expands on Bayes’ work on probability and chance.

This does beg an interesting question:  If the relationship between Bayes and Price was one of religious conviction, and if Price was a well respected minister of an important nonconformist place of worship, why would Price focus so much energy on the mathematics of chance and probability?  Keep in mind that, in the 1700s, work on probability and chance was in its relative infancy and this topic had traditionally been of interest to those who gambled and wished to have an edge over their confreres.  Therefore, it could have been viewed as unseemly for a nonconformist minister in the early stages of his career to dabble with the underpinnings of games of chance.

Price, however, had a more loftier goal in mind.

23.  The Bayesian Belief: 1. Thomas Bayes and Richard Price

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