Probability notation for Bayes' rule

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: \(\mathbb P(H).\)

  • Likelihood of evidence, conditional on the hypothesis: \(\mathbb P(e \mid H).\)

  • Posterior belief in hypothesis, after seeing evidence: \(\mathbb P(H \mid e).\)

For example, Bayes’ rule in the odds form describes the relative belief in a hypothesis \(H_1\) vs an alternative \(H_2,\) given a piece of evidence \(e,\) as follows:

$$\dfrac{\mathbb P(H_1)}{\mathbb P(H_2)} \times \dfrac{\mathbb P(e \mid H_1)}{\mathbb P(e \mid H_2)} = \dfrac{\mathbb P(H_1\mid e)}{\mathbb P(H_2\mid e)}.$$



  • Bayes' rule

    Bayes’ rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.