Bayes' theorem

English

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Bayes' theorem

Alternative forms

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's work but was apparently unaware of it.

Proper noun

Bayes ' theorem

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

,

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

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