Viewing the Kalman Filter and its variants from an optimization perspective.

1

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

2

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

3

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.

4

Derivations and discussion of linear Gaussian inverse problems.

5

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

6

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

7

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

8

An introduction to dimensionality reduction via active subspaces.

9

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

10

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

11

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

12

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.

13

I derive how a linear transformation of a Gaussian process prior influences the Gaussian process posterior, and consider some special cases.

14

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.

15

I discuss the Kalman filter from both the probabilistic and optimization perspectives.

16

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

17

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.

18

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

19

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

20

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

21