3 3 Working With Gaussians – Probability | Math for AI

Introduction

The Gaussian's superpower is closure: marginalising, conditioning, adding and linearly transforming Gaussians always gives another Gaussian, with parameters you can write down in closed form. These few formulas power Gaussian processes, Kalman filters, Bayesian linear regression and the noising process in diffusion models.

Throughout, split a joint Gaussian into two blocks:

p(\mathbf{x}, \mathbf{y}) = \mathcal{N}\!\left(\begin{bmatrix}\boldsymbol{\mu}_x\\ \boldsymbol{\mu}_y\end{bmatrix},\; \begin{bmatrix}\Sigma_{xx} & \Sigma_{xy}\\ \Sigma_{yx} & \Sigma_{yy}\end{bmatrix}\right)

Introduction

Discussion & Community