# Téoréma Bayes

Téorema Bayes mangrupa hasil dina tiori probabiliti, which gives the conditional probability distribution of a variabel acak A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone.

In the context of Bayesian probability theory and statistical inference, the marginal probability distribution of A alone is usually called the prior probability distribution or simply the prior. The conditional distribution of A given the "data" B is called the posterior probability distribution or just the posterior.

As a mathematical theorem, Bayes' théorem is valid regardless of whether one adopts a frequentist or a Bayesian interpretation of probability. However, there is disagreement as to what kinds of variables can be substituted for A and B in the théorem; this topic is tréated at gréater length in the articles on Bayesian probability and frequentist probability.

## Historical remarks

Bayes' théorem is named after the Reverend Thomas Bayes (1702–61). Bayes worked on the problem of computing a distribution for the paraméter of a binomial distribution (to use modérn terminology); his work was edited and presented posthumously (1763) by his friend Richard Price, in An Essay towards solving a Problem in the Doctrine of Chances. Bayes' results were replicated and extended by Laplace in an essay of 1774, who apparently was not aware of Bayes' work.

One of Bayes' results (Proposition 5) gives a simple description of conditional probability, and shows that it does not depend on the order in which things occur:

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, the probability I am right [i.e. the conditional probability of the first event being true given that the second has happened] is P/b.

The main result (Proposition 9 in the essay) derived by Bayes is the following: assuming a uniform distribution for the prior distribution of the binomial paraméter p, the probability that p is between two values a and b is

${\displaystyle {\frac {\int _{a}^{b}{\begin{pmatrix}n+m\\m\end{pmatrix}}p^{m}(1-p)^{n}\,dp}{\int _{0}^{1}{\begin{pmatrix}n+m\\m\end{pmatrix}}p^{m}(1-p)^{n}\,dp}}}$

where m is the number of observed successes and n the number of observed failures. His preliminary results, in particular Propositions 3, 4, and 5, imply the result now called Bayes' Théorem (as described below), but it does not appéar that Bayes himself emphasized or focused on that result.

What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the paraméter p. That is, not only can one compute probabilities for experimental outcomes, but also for the paraméter which governs them, and the same algebra is used to maké inferences of either kind. Interestingly, Bayes actually states his question in a way that might maké the idéa of assigning a probability distribution to a paraméter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial paraméter p depend on a random event, he cleverly escapes a philosophical quagmire that he most likely was not even aware was an issue.

Other Languages
aragonés: Teorema de Bayes
العربية: مبرهنة بايز
asturianu: Teorema de Bayes
беларуская: Тэарэма Баеса
беларуская (тарашкевіца)‎: Тэарэма Баеса
български: Теорема на Бейс
čeština: Bayesova věta
Ελληνικά: Θεώρημα Μπέυζ
فارسی: قضیه بیز
Gaeilge: Teoirim Bayes
עברית: חוק בייס
magyar: Bayes-tétel
Bahasa Indonesia: Teorema Bayes
íslenska: Formúla Bayes
한국어: 베이즈 정리
lietuvių: Bajeso teorema
Nederlands: Theorema van Bayes
Piemontèis: Fórmola ëd Bayes
português: Teorema de Bayes
Simple English: Bayes' theorem
српски / srpski: Бајесова теорема
svenska: Bayes sats
Türkçe: Bayes teoremi
українська: Теорема Баєса
Tiếng Việt: Định lý Bayes