[summary: Bayes' rule relates prior belief and the likelihood of evidence to posterior belief.

These quantities are often denoted using conditional probabilities:

- Prior belief in hypothesis: $~$\mathbb P(H).$~$
- Likelihood of evidence, conditional on hypothesis: $~$\mathbb P(e \mid H).$~$
- Posterior belief: $~$\mathbb P(H \mid e).$~$ ]

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:

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

## Comments

Nate Soares

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.