An Essay towards solving a Problem in the Doctrine of Chances
Bayes · Price
Summary and citation · read the original at the source
In one page
Thomas Bayes died in 1761, and his friend Richard Price found this essay among his papers, judged that it well deserved to be preserved, and had it read to the Royal Society on 23 December 1763. Its problem fits in a sentence: given how many times an unknown event has happened and how many times it has failed, what is the chance that the probability of its happening in a single trial lies somewhere between any two values you care to name? Everyone before Bayes could work forwards — knowing the chance, predict the counts. Bayes works backwards, from the counts to the chance, and Price says plainly why that matters: it is what a sure foundation for all our reasonings about past facts, and about what is likely to come, would have to rest on. The rule that carries his name is Proposition 5, and it takes four lines.
Why it matters hereChapter 1 opens on this formula because the whole site runs on it. Every document in the library is filed by how far the evidence should move you, and this is where that idea was first written down — along with the question it does not fully settle, which is what you were entitled to believe before the evidence arrived.
What it claims
01The problem the essay sets is this: given the number of times in which an unknown event has happened and failed, find the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.Statement of the Problem, p. 376
Settled physics02Proposition 5 is the rule now called Bayes' theorem: if there are two subsequent events, and you know the probability of the second and the probability of both together, then on discovering that the second event has happened, the probability that the first happened too is the probability of both together divided by the probability of the second.Proposition 5, p. 381
Settled physics03Proposition 6 gives the companion rule: the probability that several independent events all happen is a ratio compounded of the probabilities of each, since the probability of one is not altered by the happening or failing of the rest.Proposition 6, pp. 382–383
Settled physics04Price frames the stakes in his covering letter: the problem is by no means merely a curious speculation in the doctrine of chances, but must be solved in order to have a sure foundation for all our reasonings concerning past facts and what is likely to be hereafter — because common sense cannot tell us in what degree repeated experiments confirm a conclusion.Price's covering letter, pp. 371–372
Settled physics05De Moivre, after Bernoulli, had already shown how to find the probability that the observed proportion of successes in many trials falls close to the underlying chance; what Price says no one had done is the converse — deducing the unknown chance from the trials actually observed, which is what this essay solves.Price's covering letter, pp. 372–373
Settled physics06In the scholium Bayes argues that for an event about which nothing at all is known before any trials, there is no reason to think it should happen any one possible number of times rather than another in a given number of trials, and takes that as licence to apply the same rule — the assumption known ever since as the choice of prior, and the point on which every application of the rule still has to be argued.Scholium, pp. 392–394
What to watch
The way in
https://doi.org/10.1098/rstl.1763.0053Philosophical Transactions of the Royal Society 53, 370; communicated by Richard Price in a letter to John Canton and read on 23 December 1763. The essay is long out of copyright and in the public domain, and a scan of the original printing is free to read through the Royal Society and the Internet Archive; this sheet carries the summary and citation, and the original is the place to read the mathematics.
How to cite it
Bayes, Price (1763) An Essay towards solving a Problem in the Doctrine of Chances. doi:10.1098/rstl.1763.0053
Where it sits in the curriculum