Many thanks to Gregory Gundersen for the blog theme.
03 November 2024
Combining and splitting models in a Bayesian framework.
1Optimization Viewpoints on Kalman Methodology
31 October 2024
Viewing the Kalman Filter and its variants from an optimization perspective.
2Ensemble Kalman Methodology for Inverse Problems
05 October 2024
Particle-based Kalman methods for the approximate solution of Bayesian inverse problems.
328 September 2024
Motivated by Bayesian inference, I introduce the pseudo-marginal approach to MCMC and then discuss why is works from a more generic perspective.
430 July 2024
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.
5Linear Gaussian Inverse Problems
03 July 2024
Derivations and discussion of linear Gaussian inverse problems.
627 June 2024
Introduction and basic properties, kernel ridge regression, Gaussian processes.
7Basis Expansions for Black-Box Function Emulation
25 June 2024
A discussion of the popular output dimensionality strategy for emulating multi-output functions.
8A Few Different Adaptive Metropolis Schemes
10 June 2024
Adaptively updating a Gaussian proposal covariance for Random Walk Metropolis-Hastings samplers.
930 May 2024
An introduction to dimensionality reduction via active subspaces.
10A fairly deep dive into Gaussian measures in finitely many dimensions. The next step in building up to the infinite-dimensional case.
11Gaussian Measures, Part 1 - The Univariate Case
16 May 2024
A brief introduction to Gaussian measures in one dimension, serving to provide the setup for an extension to multiple, and eventually infinite, dimensions.
12An Introduction to Scattered Data Approximation
24 March 2024
I summarize the first chapter of Holger Wendland's book "Scattered Data Approximation", which I augment with some background on polynomial interpolation and splines.
13Generalizing the Outer Product to Hilbert Space
18 March 2024
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.
14Linearly Transforming Gaussian Process Priors
06 March 2024
I derive how a linear transformation of a Gaussian process prior influences the Gaussian process posterior, and consider some special cases.
15Approximating Nonlinear Functions of Gaussians, Part I - Linearized Kalman Filter Extensions
15 February 2024
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.
16The Kalman Filter - A Few Different Perspectives
15 February 2024
I discuss the Kalman filter from both the probabilistic and optimization perspectives.
17An Introduction to Bayesian Filtering and Smoothing
29 January 2024
I provide an overview of the general framework for statistical filtering, viewed from the Bayesian perspective.
18The Measure-Theoretic Context of Bayes' Rule
28 January 2024
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.
19Gaussian Process Priors, Specification and Parameter Estimation
11 January 2024
A deep dive into hyperparameter specifications for GP mean and covariance functions, including both frequentist and Bayesian methods for hyperparameter estimation.
20Deriving the Metropolis-Hastings Update from the Transition Kernel
31 December 2023
Given only the Metropolis-Hastings transition kernel, I show how to recover the Metropolis-Hastings update rule.
2115 December 2023
I derive the PCA decomposition from both a minimum reconstruction error and maximum variance perspective. I also discuss a statistical interpretation of PCA.
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