nLab 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.