## Téoréma Bayes |

**Téorema Bayes** mangrupa hasil dina
*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
*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
- statement of bayes' theorem
- derivation in the discrete case
- examples
- references
- tempo oge

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

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

Afrikaans: Bayes se stelling

aragonés: Teorema de Bayes

العربية: مبرهنة بايز

беларуская: Тэарэма Баеса

беларуская (тарашкевіца): Тэарэма Баеса

български: Теорема на Бейс

català: Teorema de Bayes

čeština: Bayesova věta

dansk: Bayes' teorem

Deutsch: Satz von Bayes

Ελληνικά: Θεώρημα Μπέυζ

English: Bayes' theorem

español: Teorema de Bayes

euskara: Bayesen teorema

فارسی: قضیه بیز

suomi: Bayesin teoreema

français: Théorème de Bayes

Gaeilge: Teoirim Bayes

galego: Teorema de Bayes

עברית: חוק בייס

हिन्दी: बेय का सिद्धांत

magyar: Bayes-tétel

Bahasa Indonesia: Teorema Bayes

íslenska: Formúla Bayes

italiano: Teorema di Bayes

日本語: ベイズの定理

한국어: 베이즈 정리

lietuvių: Bajeso teorema

монгол: Байесын теорем

Nederlands: Theorema van Bayes

norsk: Bayes' teorem

polski: Twierdzenie Bayesa

Piemontèis: Fórmola ëd Bayes

português: Teorema de Bayes

română: Teorema lui Bayes

русский: Теорема Байеса

Scots: Bayes' theorem

Simple English: Bayes' theorem

српски / srpski: Бајесова теорема

svenska: Bayes sats

தமிழ்: பேயசின் தேற்றம்

Türkçe: Bayes teoremi

українська: Теорема Баєса

اردو: بےز مسئلہ اثباتی

Tiếng Việt: Định lý Bayes

吴语: Bayes定理

中文: 贝叶斯定理

粵語: 貝葉斯定理