1

Interpreting the Cholesky factor of a Gaussian precision matrix.

2

Cholesky-like decomposition with upper-triangular matrices.

3

Interpreting the inverse covariance matrix for Gaussian distributions.

4

Interpreting and optimizing the lengthscale and marginal variance parameters of a Gaussian covariance function.

5

Interpreting the Cholesky factor of a Gaussian covariance matrix.

6

7

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

8

Combining and splitting models in a Bayesian framework.

9

Particle-based Kalman methods for the approximate solution of Bayesian inverse problems.

10

Motivated by Bayesian inference, I introduce the pseudo-marginal approach to MCMC and then discuss why is works from a more generic perspective.

11

I introduce the ensemble Kalman filter as a Monte Carlo approximation to the Kalman filter in the linear Gaussian state space setting, then discuss how is applied as an approximation even when these assumptions don't hold.

12

Derivations and discussion of linear Gaussian inverse problems.

13

Introduction and basic properties, kernel ridge regression, Gaussian processes.

14

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

15

Adaptively updating a Gaussian proposal covariance for Random Walk Metropolis-Hastings samplers.

16

An introduction to dimensionality reduction via active subspaces.

17

Gaussian measures in finite dimensions; i.e., multivariate Gaussians/Gaussian vectors.

18

A brief introduction to Gaussian measures in one dimension, serving to provide the setup for an extension to multiple, and eventually infinite, dimensions.

19

I summarize the first chapter of Holger Wendland's book "Scattered Data Approximation", which I augment with some background on polynomial interpolation and splines.

20

I briefly discuss how the outer product, familiar in Euclidean space, can be viewed as a linear operator which can readily be generalized to Hilbert space.

21

I derive generalizations of the standard Gaussian conditioning identities whereby components of the Gaussian vector are subject to linear maps, and highlight applications to Gaussian processes.

22

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.

23

I discuss the Kalman filter from both the probabilistic and optimization perspectives, and provide multiple derivations of the Kalman update.

24

I provide an overview of the general framework for statistical filtering, viewed from the Bayesian perspective.

25

I describe Bayes' rule in a measure-theoretic context, explain how it can be viewed as a non-linear operator on probability measures, and detail applications to Bayesian inverse problems.

26

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

27

Given only the Metropolis-Hastings transition kernel, I show how to recover the Metropolis-Hastings update rule.

28

I derive the PCA decomposition from both a minimum reconstruction error and maximum variance perspective. I also discuss a statistical interpretation of PCA.

29