The interactive film is in the book's Companion

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The von Mises-Fisher Distribution: A Gaussian for Directions

Chapter 13 · Mathematics for AI and Machine Learning · Advanced · 1:35

An interactive live film: exact vMF samples on the unit sphere gather around the mean direction as kappa grows; equal-density circles shrink, and the mean E[x] = A₃(κ)μ grows inside the sphere. Aim μ and set κ so the mean lands on a target.

Narration

Directions in three dimensional space live on the unit sphere: normalized embeddings, wind directions, orientations.

The von Mises Fisher distribution puts a density on the sphere: a normalizing constant times the exponential of kappa times the inner product of the mean direction mu with x. The amber arrow is mu.

The rest of the narration is in the book's Companion.

More from Chapter 13

Interactive1:35Normalizing Flows: Chaining Invertible MapsChapter 13Preview1:38Independence as an Outer ProductChapter 13Preview1:40Conditional Gaussian: A Slice of the BellChapter 13Preview1:22Random Fourier FeaturesChapter 13Preview1:46Change of Variables: Same Probability, Different AreaChapter 13Preview1:42The Dirichlet Distribution: A Density on the SimplexChapter 13Preview1:14Gaussian Mixture Model: Multimodal TerrainChapter 13Preview1:41Whitening: PCA and ZCAChapter 13Preview1:36Beta and Bernoulli: Updating with Pseudo-CountsChapter 13Preview1:43ROC Curve and AUC: the Mass Above the DiagonalChapter 13Preview1:40Hoeffding's Inequality: How Fast a Sample Mean SettlesChapter 13Preview1:25Temperature Scaling: Calibrating a SoftmaxChapter 13Preview1:34Mahalanobis Distance: The Ellipsoids of a GaussianChapter 13Preview2:09Random Directions Are Nearly OrthogonalChapter 13