1

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

2

Combining and splitting models in a Bayesian framework.

3

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

4

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

5

6

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.

7

Derivations and discussion of linear Gaussian inverse problems.

8

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

9

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

10

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

11

An introduction to dimensionality reduction via active subspaces.

12

A fairly deep dive into Gaussian measures in finitely many dimensions. The next step in building up to the infinite-dimensional case.

13

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

14

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

15

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.

16

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.

17

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.

18

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

19

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

20

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.

21

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

22

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

23

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

24