# Bayes' theorem

## English

Wikipedia has an article on:
Wikipedia

### Etymology

Named after English mathematician Thomas Bayes (1701–1761), who developed an early formulation. The modern expression of the theorem is due to Pierre-Simon Laplace, who extended Bayes' work but was apparently unaware of it.

### Proper noun

Bayes' theorem

1. (probability theory) A theorem expressed as an equation that describes the conditional probability of an event or state given prior knowledge of another event.
• 2010, Jonathan Harrington, Phonetic Analysis of Speech Corpora, page 327,
The starting point for many techniques in probabilistic classification is Bayes' theorem, which provides a way of relating evidence to a hypothesis.
• 2011, Allen Downey, Think Stats, O'Reilly, page 56,
Bayes's theorem is a relationship between the conditional probabilities of two events.
• 2013, Norman Fenton, Martin Neil, Risk Assessment and Decision Analysis with Bayesian Networks, Taylor & Francis (CRC Press), page 131,
We have now seen how Bayes' theorem enables us to correctly update a prior probability for some unknown event when we see evidence about the event.

#### Usage notes

The theorem is stated mathematically as:

${\displaystyle \displaystyle P(A\mid B)={\frac {P(B\mid A)\,P(A)}{P(B)}}}$,

where ${\displaystyle \textstyle A}$ and ${\displaystyle \textstyle B}$ are events with ${\displaystyle \textstyle P(B)\neq 0}$, and

• ${\displaystyle \textstyle P(A)}$ and ${\displaystyle \textstyle P(B)}$ are the marginal probabilities of observing ${\displaystyle \textstyle A}$ and ${\displaystyle \textstyle B}$ without regard to each other.
• The conditional probability ${\displaystyle \textstyle P(A\mid B)}$ is the probability of observing event ${\displaystyle \textstyle A}$ given that ${\displaystyle \textstyle B}$ is true.
• Similarly, ${\displaystyle \textstyle P(B\mid A)}$ is the probability of observing event ${\displaystyle \textstyle B}$ given that ${\displaystyle \textstyle A}$ is true.