# 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:

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

Children:

Parents:

• Bayes' rule

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

• I suggest making it explicit that $$P$$ is a distribution over a (possibly infinite) set of variables (or propositions naming symbols, or whatever your preferred formalization is), and that $$P(x)$$ is shorthand for $$P(X=x)$$ when $$X$$ is unambiguous. This is one of those things that I had to figure out myself, which had confused me historically in my youth, and led me to think that all the $$P$$ notation was probably informal argument rather than formal math.