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)