Probability notation for Bayes' rule

[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.