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.

Nate Soares 3 Mar 2016 18:03 UTC
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.