# 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:

- Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.

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.