Andrew G. Roberts

The Reversed Cholesky Decomposition

Linear-Algebra

Cholesky-like decomposition with upper-triangular matrices.

Regularized Least Squares with Singular Prior

Statistics

Data-Assimilation

Optimization

Inverse-Problem

Solving the regularized least squares optimization problem when the prior covariance matrix is not positive definite.

Nonlinear Least Squares

Statistics

Optimization

Inverse-Problem

Gauss-Newton, Levenberg-Marquardt

Ensemble Kalman Methods for Solving Inverse Problems: An Introduction

Inverse-Problem

Data-Assimilation

Optimization

Sampling

Computational Statistics

EKI

The first post in a series on using Ensemble Kalman methods to approximately solve Bayesian inverse problems.

Basis Expansions for Black-Box Function Emulation

Gaussian-Process

A discussion of the popular output dimensionality strategy for emulating multi-output functions.

Approximating Nonlinear Functions of Gaussians

And the Extended Kalman Filter

Data-Assimilation

I discuss the generic problem of approximating the distribution resulting from a non-linear transformation of a Gaussian random variable, and then show how this leads to extensions of the Kalman filter which yield approximate filtering algorithms in the non-linear setting.

Introduction to Gaussian Process Priors and Hyperparameter Estimation

Statistics

Gaussian-Process

kernel-methods

A deep dive into hyperparameter specifications for GP mean and covariance functions, including both frequentist and Bayesian methods for hyperparameter estimation.

Deriving the Metropolis-Hastings Update from the Transition Kernel

MCMC

Probability

Computational Statistics

The Metropolis-Hastings (MH) Markov Chain Monte Carlo (MCMC) method is typically introduced in the form of a practical algorithm. In a more theoretically-oriented context…

Principal Components Analysis

Statistics

Linear-Algebra

A deep dive into PCA.