Discriminant analysis

Discriminant analysis constructs statistical rules distinguishing labelled classes from their measured feature values. A common generative approach models the class-conditional probability distribution and applies Bayes' theorem to compare posterior class probabilities. Linear discriminant analysis uses multivariate normal distributions with a shared covariance matrix; allowing a different covariance matrix for each class gives quadratic discriminant analysis.^[ref]

For two equally likely one-dimensional classes, suppose $X\mid C_1\sim N(0,1)$ and $X\mid C_2\sim N(2,1)$. Their log-likelihood ratio isChoosing the larger posterior therefore selects $C_2$ for $x>1$, with a tie at one. Different class probabilities move this threshold. Estimating means and covariance from training data supplies a fitted rule, while independent evaluation tests whether its assumptions generalize.