= Discriminant analysis
<Discriminant analysis> constructs statistical rules distinguishing labelled classes from their measured <feature (machine learning)> 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>.<Linear and Quadratic Discriminant Analysis in the scikit-learn documentation>[§1.2.2, probabilistic model and shared-covariance simplification]{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 is
$$
\log\frac{p(x\mid C_2)}{p(x\mid C_1)}=-\frac{(x-2)^2}{2}+\frac{x^2}{2}=2x-2.
$$
Choosing 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.
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