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: $P\left(H\right).$
• Likelihood of evidence, conditional on the hypothesis: $P(e\mid H).$
• Posterior belief in hypothesis, after seeing evidence: $P(H\mid e).$

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:

$$\frac{P\left(H_1\right)}{P\left(H_2\right)}\times \frac{P(e\mid H_1)}{P(e\mid H_2)}=\frac{P(H_1\mid e)}{P(H_2\mid e)}.$$