bayes-rule

Canonically, for events $x$ and $y$, we have the definition:

$p(x \vert y) = \frac{p(y \vert x)p(x)}{p(y)}$

where $p(y)$ is the [[evidence]], $p(x)$ is the [[prior]], $p(x \vert y)$ is the [[posterior distribution]], and $p(y \vert x)$ is the [[conditional-distribution]] known as the likelihood.

Another way written, as seen in Josh Speagle's slides, is to write it as:

$\text{Pr}(\Theta \vert D,M) = \frac{\text{Pr}(D \vert \Theta, M)\text{Pr}(\Theta \vert M)}{\text{Pr}(D \vert M)}$

where aspects are more clearly defined as $M$ model dependent (such as the prior), for model parameters $\Theta$ and data $D$.