2026-01-20

2026-01-14 Geometry of Data Notes

Information Geometry

Parameters live on a manifold

Example

Covariance Matrix of a 2D Gaussian Distribution

$$ \Sigma = \begin{bmatrix} \mathrm{var}(X) & \mathrm{cov}(X, Y)\ \mathrm{cov}(X, Y) & \mathrm{Var}(Y) \end{bmatrix} $$

Renaming the variables to something simple $$ \Sigma = \begin{bmatrix} a & b \\ b & c \end{bmatrix} $$

Notice the constraints on this matrix

• We know that $a, c$ are positive since they are variances