Wishart Bartlett Construction

The Wishart distribution is a family of probability distributions defined over symmetric, nonnegative-definite matrices. It represents the distribution of the sample covariance matrix when drawing independent samples from a multivariate normal distribution. The Bartlett decomposition provides an efficient and insightful way to construct a random Wishart matrix from independent, scalar random variables without directly simulating high-dimensional vectors and forming their sample covariance.

Experiment

We can observe how these parameters independently affect the matrix structure and geometry by interacting with the module.

1. Change the Scale to "Anisotropic". The ellipses stretch along the vertical axis, and the expected trace jumps from 6.0 (for Correlated) to 15.0, reflecting Σ = [[1, 0], [0, 4]].
2. Increase the Degrees to 5. Notice how the expected trace and determinant increase proportionally, but the matrix W remains unchanged since the primitive values q₁, q₂, and z are fixed.
3. Set Normal value to 0. This removes the off-diagonal correlation introduced by z, which changes the covariance ellipse of W to perfectly align with the axes when the scale is "Identity".