# Bayes Decision Theory
Goal: predict output class $\mathcal{C}$ from measurements $\text{x}$ by minimizing the probability of misclassification

>[!tip] Main Equation:
>$$p(X,Y)=\frac{p(X|Y)p(Y)}{p(X)}$$


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$posterior = \frac{likelihood \cdot prior}{normalization factor}$


==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:
$$
\mathcal{N}(\text{x}|\mu, \Sigma) = \frac{1}{(2\pi)^{D/2}|\Sigma|^{1/2}}\exp\left(-\frac{1}{2} (\text{x}-\mu)^\top \Sigma^{-1}(\text{x}-\mu) \right)
$$