Linear Discriminant Analysis (LDA)
A generative linear classifier that models each class as a multivariate Gaussian with a class-specific mean but a single shared (pooled) covariance matrix. Under these assumptions the Bayes-optimal decision boundary is linear, and the discriminant projects a feature vector onto the direction w = Σ^{-1}(μ1 − μ0). Equivalently, Fisher's criterion chooses the projection that maximizes the ratio of between-class scatter to within-class scatter.
LDA dominates practical BCI because, given few well-designed features (band powers, CSP components, ERP amplitudes) and limited calibration data, its strong bias (a linear boundary and shared covariance) yields low variance, and it trains in closed form with no hyperparameters. Its weakness is the covariance inverse: when the number of features approaches the number of trials the sample covariance is ill-conditioned and LDA overfits catastrophically, which is the direct motivation for shrinkage.
LDA is often conflated with logistic regression; both give linear boundaries, but LDA is generative (it models p(x|y) and derives the boundary from class means and covariance) while logistic regression is discriminative (it models p(y|x) directly) and is more robust when the Gaussian assumption fails.