---
title: "Bayes Theorem"
type: concept
source: https://www.jemoka.com/posts/kbhbayes_theorem/
confidence: high
status: active
---

\begin{align} p(x\mid y) = \frac{p(y \mid x) p(x)}{p(y)} \end{align}
this is a direct result of the probability chain rule.
Typically, we name \(p(y|x)\) the &ldquo;likelihood&rdquo;, \(p(x)\) the &ldquo;prior&rdquo;.
Better normalization What if you don&rsquo;t fully know \(p(y)\), say it was parameterized over \(x\)?
\begin{align} p(x|y) &amp;= \frac{p(y|x) \cdot p(x)}{p(y)} \\ &amp;= \frac{p(y|x) \cdot p(x)}{\sum_{X_{i}} p(y|X_{i})} \end{align}
just apply law of total probability! taad
