Editing Thomas Bayes (section)

== Bayes' theorem ==
{{Main article|Bayes' theorem}}

Bayes's solution to a problem of [[inverse probability]] was presented in ''[[An Essay Towards Solving a Problem in the Doctrine of Chances]]'', which was read to the [[Royal Society]] in 1763 after Bayes's death. [[Richard Price]] shepherded the work through this presentation and its publication in the ''Philosophical Transactions of the Royal Society of London'' the following year.<ref>{{cite journal|author=Bayes, Thomas|doi=10.1098/rstl.1763.0053|title=An Essay Towards Solving a Problem in the Doctrine of Chances|journal=Philosophical Transactions|volume=53|year=1763|pages=370–418|s2cid=186213794|doi-access=free}}</ref> This was an argument for using a uniform prior distribution for a binomial parameter and not merely a general postulate.<ref>Edwards, A. W. G. [https://www.jstor.org/pss/4615697 "Commentary on the Arguments of Thomas Bayes,"] ''Scandinavian Journal of Statistics'', Vol. 5, No. 2 (1978), pp. 116–118; retrieved 6 August 2011</ref> This essay gives the following theorem (stated here in present-day terminology).
<blockquote> Suppose a quantity ''R'' is [[uniform distribution (continuous)|uniformly distributed]] between 0 and&nbsp;1. Suppose each of ''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub> is equal to either 1 or 0 and the [[conditional probability]] that any of them is equal to&nbsp;1, given the value of ''R'', is&nbsp;''R''. Suppose they are [[conditional independence|conditionally independent]] given the value of&nbsp;''R''. Then the conditional probability distribution of&nbsp;''R'', given the values of ''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub>, is
: <math>
\frac {(n+1)!}{S!(n-S)!} r^S (1-r)^{n-S} \, dr \quad \text{for }0\le r\le 1, \text{ where } S=X_1+\cdots+X_n.
</math></blockquote>
Thus, for example,
: <math> \Pr(R \le r_0 \mid X_1,\ldots,X_n) = \frac{(n+1)!}{S!(n-S)!} \int_0^{r_0} r^S (1-r)^{n-S} \, dr.
</math>

This is a special case of the [[Bayes' theorem]].

In the first decades of the eighteenth century, many problems concerning the probability of certain events, given specified conditions, were solved. For example: given a specified number of white and black balls in an urn, what is the probability of drawing a black ball? Or the converse: given that one or more balls has been drawn, what can be said about the number of white and black balls in the urn? These are sometimes called "[[inverse probability]]" problems.

Bayes's ''Essay'' contains his solution to a similar problem posed by [[Abraham de Moivre]], author of ''[[The Doctrine of Chances]]'' (1718).

In addition, a paper by Bayes on [[asymptotic series]] was published posthumously.