An Essay towards solving a Problem in the Doctrine of Chances
How a single new fact should change your mind — by exactly how much.
A new fact arrives. By exactly how much should it change your mind?
Reasoning backwards from evidence
Most of us are taught probability forwards: if a bag holds 7 red marbles and 3 blue, the chance of drawing red is 7 in 10. Bayes asked the harder, more useful question — the backwards one. You don't know what's in the bag; you've only seen the marbles you've drawn. From those draws, what can you now believe about the bag? His essay showed how to turn observations into a precise, updatable belief about the hidden cause behind them.
An essay found among a dead man's papers
Thomas Bayes was an English Presbyterian minister and amateur mathematician who published almost nothing in his lifetime. When he died in 1761, his friend Richard Price was asked to look through his papers, and there found an unfinished essay on chance. Price recognised its worth, edited it, added worked examples, and read it to the Royal Society in 1763 — two years after its author was dead. 'I now send you an essay which I have found among the papers of our deceased friend Mr. Bayes,' he wrote. For nearly a century the idea drew little attention, until the great French mathematician Pierre-Simon Laplace, working independently, developed it into a full theory and showed its power across science.
Why it matters
Bayes's rule is the mathematics of changing your mind well. It says precisely how much a new piece of evidence should shift your confidence: start with what you believed before (the prior), weigh it by how well the evidence fits each possibility (the likelihood), and you arrive at a revised belief (the posterior). Get the prior wrong and you can be badly misled — which is why a positive result on a test for a rare disease is so often a false alarm, and why thinking in base rates is one of the most practical lessons probability has to teach.
Tuning the dial
Imagine a dial marked with every possible 'true rate' of some event, from never to always. Before any evidence, your belief is spread evenly across the whole dial. Each observation nudges the needle: a success pushes belief toward 'more often,' a failure toward 'less often.' At first the belief is a broad blur covering most of the dial; with more data it tightens into a sharp peak around the rate the world keeps showing you. Bayes's essay is the exact rule for how far each observation moves the needle.
Where it sits
Bayes stands at the headwaters of statistics and of modern machine reasoning. The forward probability he inverted had been worked out by Pascal, Fermat and Jacob Bernoulli; Laplace would turn Bayes's special case into a general law; and two centuries later the idea would merge with Claude Shannon's information theory — where learning a fact literally means cutting your uncertainty roughly in half — and with the statistical learning behind systems like AlexNet and the Transformer. Every time a machine 'updates on evidence,' it is running a descendant of this essay.
Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.
The purpose I mean is, to shew what reason we have for believing that there are in the constitution of things fixt laws according to which things happen, and that, therefore, the frame of the world must be the effect of the wisdom and power of an intelligent cause; and thus to confirm the argument taken from final causes for the existence of the Deity.