# nLab Bayes rule

Bayes rule

### Context

#### Measure and probability theory

measure theory

probability theory

# Bayes rule

## Statement

In probability theory, the Bayes Rule (or Bayes's Law, Bayes' Theorem, or another permutation) is the statement that the conditional probabilities $P(H\vert E)$ for an event $H$, assuming an event $E$ is related to the condition probability $P(E\vert H)$ of $E$ assuming $H$, and the plain probabilities $P(H)$ and $P(E)$ for $H$ and $E$ separately, by

$P(H|E) \;=\; \frac{P(E|H) P(H)} {P(E)} \,.$

This follows directly from the defining formula $P(A|B) = P(A \wedge B)/P(B)$ for conditional probability. The rule may also be written in the expanded form

$P(H|E) = \frac{P(E|H) P(H)} {P(E|H) P(H) - P(E|\neg{H}) P(H) + P(E|\neg{H})} ,$

which additionally uses some of the axioms of probability, or somewhere in between these two forms.

As a theorem, it is quite trivial; the point is in its application as a rule for updating the probability of some hypothesis ($H$) on the basis of some evidence ($E$) (which is key to Bayesianism), using the prior probability of the hypothesis before the evidence is obtained ($P(H)$) and (in the expanded form) the conditional probabilities of obtaining that evidence in the situation where the hypothesis is true ($P(E|H)$) and in the situation where the hypothesis is false ($P(E|\neg{H})$). (Ideally, the last two can be determined on a purely theoretical basis, but since the probability of $E$ usually depends on other hypotheses of unknown veracity, the application is not always so simple.)

## Applications

### In quantum physics

In quantum mechanics, the collapse of the wavefunction may be seen as a generalization of Bayes's Rule to quantum probability theory. This is key to the Bayesian interpretation of quantum mechanics.

Last revised on September 12, 2018 at 15:09:37. See the history of this page for a list of all contributions to it.