Probability notation for Bayes' rule

Written by Eliezer Yudkowsky, Nate Soares, Alonzo Church last updated

Bayes' rule relates prior belief and the likelihood of evidence to posterior belief.

These quantities are often written using conditional probabilities:

  • Prior belief in the hypothesis:
  • Likelihood of evidence, conditional on the hypothesis:
  • Posterior belief in hypothesis, after seeing evidence:

For example, Bayes' rule in the odds form describes the relative belief in a hypothesis vs an alternative given a piece of evidence as follows: