Bayes decided that his goal was to learn the approximate probability of a future event he knew nothing about except its past, that is, the number of times it had occurred or failed to occur.

the law of insufficient reason (meaning that without enough data to assign specific probabilities, equal ones would suffice). Despite their venerable history, equal probabilities would become a lightning rod for complaints that Bayes was quantifying ignorance.

Next, he tried to confirm that Graunt was correct about the probability of a boy’s birth being larger than 50%. He was building the foundation of the modern theory of testing statistical hypotheses.

Laplace continued his research throughout France’s political upheavals. In 1810 he announced the central limit theorem, one of the great scientific and statistical discoveries of all time. It asserts that, with some exceptions, any average of a large number of similar terms will have a normal, bell-shaped distribution.

He transformed probability from a gambler’s measure of frequency into a measure of informed belief.

At its heart, Bayes runs counter to the deeply held conviction that modern science requires objectivity and precision. Bayes is a measure of belief. And it says that we can learn even from missing and inadequate data, from approximations, and from ignorance.

On its face Bayes’ rule is a simple, one-line theorem: by updating our initial belief about something with objective new information, we get a new and improved belief. To its adherents, this is an elegant statement about learning from experience.

Because we can seldom be certain that a particular cause will have a particular effect, we must be content with finding only probable causes and probable effects.

By rights, Bayes’ rule should be named for someone else: a Frenchman, Pierre Simon Laplace, one of the most powerful mathematicians and scientists in history. To deal with an unprecedented torrent of data, Laplace discovered the rule on his own in 1774.