The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

Get the book

Discretization Changes the Stationary Distribution

Chapter 21 · Mathematics for AI and Machine Learning · Advanced · 1:52

An interactive live film: the unadjusted Langevin chain for a standard Gaussian settles at variance 1/(1 - eta/2) instead of 1; a step-size slider moves a wall over the stationary densities, with a fixed-seed chain's histogram, while the KL divergence to the target decays exponentially to a floor above zero.

Narration

The target is the standard Gaussian, the blue curve on the wall. The floor runs along x and along the step size eta, from zero to one.

The algorithm runs the discretized chain: the next x is one minus eta times x, plus the square root of two eta times fresh noise.

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

More from Chapter 21

Interactive1:35Fokker-Planck: A Random Cloud, a Deterministic DensityChapter 21Preview1:29Detailed Balance: Stationary Is Not the Same as StillChapter 21Preview1:28Forward Diffusion and the Heat Equation: All Distributions Become GaussianChapter 21Preview1:27Guidance Does Not Sample the Sharpened DistributionChapter 21Preview1:50Metastability and the Kramers Escape RateChapter 21Preview1:26Reverse Time: Why We Must Learn the ScoreChapter 21Preview1:34Unification: SDEs, Probability Flow ODEs, and Fokker-Planck DynamicsChapter 21Preview1:29Stein Variational Gradient Descent: Pull and PushChapter 21