Blog

[Derivation Scribbles] Variational Inference and the ELBO

Derivation Scribbles

Variational inference to approximate Bayesian posteriors and the evidence lower bound.

Aug 1, 2026
Rui-Yang Zhang
 

Some Properties of Expected Information Gain

Active Learning
Gaussian Process

Predictive and parameter EIG, relationship with uncertainty sampling and BALD.

Apr 2, 2026
Rui-Yang Zhang
 

[Derivation Scribbles] Exact Solution to a Multivariate OU Process

Derivation Scribbles
Stochastic Calculus

Exact solution to a multivarite Ornstein-Uhlenbeck process.

Mar 12, 2026
Rui-Yang Zhang

[Derivation Scribbles] Probabilistic ODE Solvers

Derivation Scribbles
Data Assimilation
Gaussian Process

Probabilistic Numerics ODE Solver

Feb 26, 2026
Rui-Yang Zhang

[Derivation Scribbles] Fourier Transform and FFT

Derivation Scribbles

Describes the Fourier transform, mentions the sampling theorem that connects discrete and continuous time Fourier transform, and presents the fast Fourier transform that efficiently applies the discrete Fourier transform to a uniformly sampled sequence.

Oct 12, 2025
Rui-Yang Zhang

Approximate GP Posterior Sampling via the Matheron Rule

Gaussian Process

The Matheron rule of conditional multivariate normal distribution offers a new way to approximately sample from a posterior Gaussian process.

Oct 9, 2025
Rui-Yang Zhang

Monte Carlo, Antithetic, and Quasi Monte Carlo

Sampling

Basics of Monte Carlo, Antithetic, Quasi Monte Carlo, Randomised Quasi Monte Carlo.

Aug 11, 2025
Rui-Yang Zhang

Upscale a Brownian Motion Path

Gaussian Process
Sampling
Stochastic Calculus

For a Brownian motion path observed at fixed locations, upscale the resolution using conditioning using the fact that a Brownian motion is a Gaussian process.

Jul 30, 2025
Rui-Yang Zhang

[Reading Notes] Deep Learning is Not So Mysterious or Different

Deep Learning
Reading Notes

Notes on reading Andrew Gordon Wilson’s ICML 2025 paper Deep Learning is Not So Mysterious or Different.

Jul 11, 2025
Rui-Yang Zhang

Toy Bayesian Data Assimilation

Data Assimilation

Bayesian data assimilation for a toy problem.

Jul 1, 2025
Rui-Yang Zhang

Identifiability Issues of Gaussian Processes

Gaussian Process

Describe the consistency and identifiability of Matérn GPs.

Jun 18, 2025
Rui-Yang Zhang

Spatial-Temporal GP (3)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. SDE Approach to Temporal GP Regression.

Mar 30, 2025
Rui-Yang Zhang

Expected Information Gain with Gaussian Process Surrogate Models

Active Learning
Gaussian Process

Computations and derivations of the expected information gain utility function of active learning when the surrogate model is a conjugate Gaussian process.

Feb 7, 2025
Rui-Yang Zhang

Summary of a Bivariate Gaussian Covariance Matrix

Active Learning
Gaussian Process

Various ways one could summarise a bivariate Gaussian’s covariance matrix.

Feb 3, 2025
Rui-Yang Zhang

[Derivation Scribbles] 3D-Var and 4D-Var

Data Assimilation
Derivation Scribbles

Derivations for 3D-Var and 4D-Var.

Dec 3, 2024
Rui-Yang Zhang

[Derivation Scribbles] Kalman Filter and Ensemble Kalman Filter

Data Assimilation
Derivation Scribbles

Derivations for Kalman filters and Ensemble Kalman Filter.

Nov 28, 2024
Rui-Yang Zhang

Spatial-Temporal GP (2)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. Exploiting the Kronecker structure of temporal GP regression with 1d space.

Oct 31, 2024
Rui-Yang Zhang

Spatial-Temporal GP (1)

Gaussian Process

A series of blog posts on spatial-temporal Gaussian processes. Temporal GP regression with 1d space.

Oct 23, 2024
Rui-Yang Zhang

[Derivation Scribbles] Basic GP Regression Formula

Gaussian Process
Derivation Scribbles

Derivations for the Gaussian process predictive distribution. Single-output GP, observe with additive Gaussian noise.

Oct 13, 2024
Rui-Yang Zhang

Why Should We Care About Gradient Flows?

Gradient Flow

Blog post on gradient flows in Euclidean and Wasserstein spaces.

Sep 13, 2024
Rui-Yang Zhang, Christopher Nemeth
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