''An Essay towards solving a Problem in the Doctrine of Chances'' is a work on the mathematical
theory of probability
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
by
Thomas Bayes, published in 1763,
[ two years after its author's death, and containing multiple amendments and additions due to his friend Richard Price. The title comes from the contemporary use of the phrase "doctrine of chances" to mean the theory of probability, which had been introduced via the title of a book by ]Abraham de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He move ...
. Contemporary reprints of the Essay carry a more specific and significant title: ''A Method of Calculating the Exact Probability of All Conclusions founded on Induction''.
The essay includes theorems of conditional probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
which form the basis of what is now called Bayes's Theorem, together with a detailed treatment of the problem of setting a prior probability
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
.
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
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
is
:
(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
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
with parameters ''k'' + 1 and ''n'' − ''k'' + 1.
Outline
Bayes's preliminary results in conditional probability (especially Propositions 3, 4 and 5) imply the truth of the theorem that is named for him. He states:''"If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event has also happened, from hence I guess that the first event has also happened, the probability I am right is P/b."''.
Symbolically, this implies (see Stigler 1982):
:
which leads to Bayes's Theorem for conditional probabilities:
:
However, it does not appear that Bayes emphasized or focused on this finding. Rather, he focused on the finding the solution to a much broader inferential problem:
:''"Given the number of times in which an unknown event has happened and failed .. Findthe chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named."''
The essay includes an example of a man trying to guess the ratio of "blanks" and "prizes" at a lottery. So far the man has watched the lottery draw ten blanks and one prize. Given these data, Bayes showed in detail how to compute the probability that the ratio of blanks to prizes is between 9:1 and 11:1 (the probability is low - about 7.7%). He went on to describe that computation after the man has watched the lottery draw twenty blanks and two prizes, forty blanks and four prizes, and so on. Finally, having drawn 10,000 blanks and 1,000 prizes, the probability reaches about 97%.
Bayes's main result (Proposition 9) is the following in modern terms:
:Assume a uniform prior distribution of the binomial parameter . After observing successes and failures,
::