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

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An introduction to dimensionality reduction via active subspaces.

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

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

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

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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.

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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.

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

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

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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.

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

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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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