Who Published Bayes' Theorem?

For the legal profession, "American Law Reports or ALR is a longstanding, highly trusted series of in-depth articles, called annotations, on specific legal issues."^\1/ But the report on "Use of Bayes' Theorem in Criminal and Civil Cases"^\2/ may be less worthy of trust than the average annotation. Historians and statisticians, at least, would be surprised at its opening sentences:

Bayes' Theorem, named for the English clergyman and scientist who published it in 1763, is a scientific principle of the likelihood ratio used to calculate conditional probabilities leading to a variety of statistical "Bayesian" methodologies. It speculates on the probability that a particular fact is true or that a particular event will occur, given our knowledge of one or more related facts or events.

There is no doubt (asymptotically speaking) that in 1763, Thomas Bayes did not publish the theorem that bears his name. He was dead. A close friend, the Reverend Richard Price, read the paper to the Royal Society of London after retrieving it from Bayes's papers and adding to it.^\3/ This expanded and modified version appeared in the Society's Philosophical Transactions.

The ALR's rendering of 18th Century history won't cause any problems for lawyers employing or confronting Bayesian analyses in court. Posthumous or otherwise, discovered first by Bayes or by someone else,^\4/ the theorem is what it is. But what will lawyers make of the idea that the theorem is "a scientific principle of the likelihood ratio" that "speculates" about conditional probability? Is the theorem mere speculation? Is it only about a likelihood ratio? No, and no.

The usual understanding is that a likelihood ratio is the data-based, objective part of the theorem and that speculation becomes an issue if and when the prior probability is not objective as well. As Judea Pearl and Dana MacKenzie explain,

Bayesian statistics gives us an objective way of combining the observed evidence with our prior knowledge (or subjective belief) to obtain a revised belief ... . Still, what frequentists could not abide was that Bayesians were allowing opinion, in the form of subjective probabilities, to intrude into the pristine kingdom of statistics. Mainstream statisticians were won over only grudgingly, when Bayesian analysis proved a superior tool for a variety of applications ... .^\5/

Fitting expert prior probabilities and likelihoods into the legal process poses some special problems for mainstream statisticians. That accommodation is the subject of continuing, if often repetitive, multidisciplinary dialog that does not fit neatly and quickly into a blog.

Notes

1. University of South Carolina School of Law, Legal Research, Analysis & Writing, https://guides.law.sc.edu/LRAWSpring/LRAW/alr.
2. Deborah F. Buckman, Annot., Use of Bayes' Theorem in Criminal and Civil Cases, 47 A.L.R.7th Art. 4 (2019).
3. Stephen M. Stigler, Richard Price, The First Bayesian, 33 Statistical Science 117 (2018), https://doi.org/10.1214/17-STS635; Martyn Hooper, Richard Price, Bayes’ Theorem, and God, Significance 36-39 (Feb. 2013), https://www.york.ac.uk/depts/maths/histstat/price.pdf.
4. Stephen M. Stigler, Who Discovered Bayes’s Theorem?, 37 Am. Stat. 290-296 (1983).
5. Judea Pearl & Dana MacKenzie, The Book of Why: The New Science of Cause and Effect 90 (2018).