23 lines
629 B
Markdown
23 lines
629 B
Markdown
# Bayes Decision Theory
|
|
Goal: predict output class $\mathcal{C}$ from measurements $\text{x}$ by minimizing the probability of misclassification
|
|
|
|
>[!tip] Main Equation:
|
|
>$$p(X,Y)=\frac{p(X|Y)p(Y)}{p(X)}$$
|
|
|
|
|
|
---
|
|
|
|
$posterior = \frac{likelihood \cdot prior}{normalization factor}$
|
|
|
|
|
|
==NOTE: Learn Normal/Gaussian Distribution by heart!==
|
|
univariate:
|
|
$$
|
|
\mathcal{N} (x|\mu, \sigma^2) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)
|
|
$$
|
|
multivariate:
|
|
$$
|
|
\mathcal{N}(\text{x}|\mu, \Sigma) = \frac{1}{(2\pi)^{D/2}|\Sigma|^{1/2}}\exp\left(-\frac{1}{2} (\text{x}-\mu)^\top \Sigma^{-1}(\text{x}-\mu) \right)
|
|
$$
|
|
|