An Essay towards solving a Problem in the Doctrine of Chances

An Essay towards solving a Problem in the Doctrine of Chances is a work on the mathematical theory of probability by the Reverend Thomas Bayes, published in 1763,[1] two years after its author's death. It included a statement of a special case of what is now called Bayes' theorem. In 18th-century English, the phrase "doctrine of chances" meant the theory of probability. It had been introduced as the title of a book by Abraham de Moivre.

Bayes supposed a sequence of independent experiments, each having as its outcome either success or failure, the probability of success being some number p between 0 and 1. But then he supposed p to be an uncertain quantity, whose probability of being in any interval between 0 and 1 is the length of the interval. In modern terms, p would be considered a random variable uniformly distributed between 0 and 1. Conditionally on the value of p, the trials resulting in success or failure are independent, but unconditionally (or "marginally") they are not. That is because if a large number of successes are observed, then p is more likely to be large, so that success on the next trial is more probable. The question Bayes addressed was: what is the conditional probability distribution of p, given the numbers of successes and failures so far observed. The answer is that its probability density function is

f(p) = \frac{(n%2B1)!}{k!(n-k)!} p^k (1-p)^{n-k}\text{ for }0\le p \le 1

(and ƒ(p) = 0 for p < 0 or p > 1) where k is the number of successes so far observed, and n is the number of trials so far observed. This is what today is called the Beta distribution with parameters k + 1 and n − k + 1.

References

  1. ^ Bayes T. (1763) "An Essay towards solving a Problem in the Doctrine of Chances". Phil. Trans., 53, 370–418. doi:10.1098/rstl.1763.0053

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