# Posterior Gaussian process samples by updating prior samples

Matheron’s other weird trick

December 3, 2020 — May 8, 2024

\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\cov}{\operatorname{Cov}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\dif}{\backslash} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Can we find a transformation that will turn a Gaussian process prior sample into a Gaussian process posterior sample. A special trick where we do GP regression by GP simulation.

The main tool is an old insight made useful for modern problems in J. T. Wilson et al. (2020) (brusque) and J. T. Wilson et al. (2021) (deep). Actioned in Ritter et al. (2021) to condition probabilistic neural nets somehow.

## 1 Matheron updates for Gaussian RVs

We start by examining a slightly different way of defining a Gaussian RV (J. T. Wilson et al. 2021), starting from the recipe for sampling:

A random vector \(\boldsymbol{x}=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}\) is said to be Gaussian if there exists a matrix \(\mathbf{L}\) and vector \(\boldsymbol{\mu}\) such that \[ \boldsymbol{x} \stackrel{\mathrm{d}}{=} \boldsymbol{\mu}+\mathbf{L} \boldsymbol{\zeta} \quad \boldsymbol{\zeta} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}) \] where \(\mathcal{N}(\mathbf{0}, \mathbf{I})\) is known as the standard version of a (multivariate) normal distribution, which is defined through its density.

This is the location-scale form of a Gaussian RV, as opposed to the canonical form which we use in Gaussian Belief Propagation. In location-scale form, a non-degenerate Gaussian RV’s distribution is given (uniquely) by its mean \(\boldsymbol{\mu}=\mathbb{E}(\boldsymbol{x})\) and its covariance \(\boldsymbol{\Sigma}=\mathbb{E}\left[(\boldsymbol{x}-\boldsymbol{\mu})(\boldsymbol{x}-\boldsymbol{\mu})^{\top}\right] .\) In this notation the density, if defined, is \[ p(\boldsymbol{x})=\mathcal{N}(\boldsymbol{x} ; \boldsymbol{\mu}, \boldsymbol{\Sigma})=\frac{1}{\sqrt{|2 \pi \boldsymbol{\Sigma}|}} \exp \left(-\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right). \]

Since \(\zeta\) has identity covariance, any matrix square root of \(\boldsymbol{\Sigma}\), such as the Cholesky factor \(\mathbf{L}\) with \(\boldsymbol{\Sigma}=\mathbf{L L}^{\top}\), may be used to draw \(\boldsymbol{x}=\boldsymbol{\mu}+\mathbf{L} \boldsymbol{\zeta}.\)

**tl;dr** we can think about drawing any Gaussian RV as transforming a standard Gaussian. So much is basic entry-level stuff. What might a rule which updates a Gaussian prior into a data-conditioned posterior look like? Like this!

We define \(\cov(a,b)=\Sigma_{a,b}\) as the covariance between two random variables (J. T. Wilson et al. 2021):

Matheron’s Update Rule: Let \(\boldsymbol{a}\) and \(\boldsymbol{b}\) be jointly Gaussian, centered random variables. Then the random variable \(\boldsymbol{a}\) conditional on \(\boldsymbol{b}=\boldsymbol{\beta}\) may be expressed as \[ (\boldsymbol{a} \mid \boldsymbol{b}=\boldsymbol{\beta}) \stackrel{\mathrm{d}}{=} \boldsymbol{a}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}}{\boldsymbol{\Sigma}}_{\boldsymbol{b}, \boldsymbol{b}}^{-1}(\boldsymbol{\beta}-\boldsymbol{b}) \] Proof: Comparing the mean and covariance on both sides immediately affirms the result \[ \begin{aligned} \mathbb{E}\left(\boldsymbol{a}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1}(\boldsymbol{\beta}-\boldsymbol{b})\right) & =\boldsymbol{\mu}_{\boldsymbol{a}}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1}\left(\boldsymbol{\beta}-\boldsymbol{\mu}_{\boldsymbol{b}}\right) \\ & =\mathbb{E}(\boldsymbol{a} \mid \boldsymbol{b}=\boldsymbol{\beta}) \end{aligned} \] \[ \begin{aligned} \operatorname{Cov}\left(\boldsymbol{a}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1}(\boldsymbol{\beta}-\boldsymbol{b})\right) &=\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{a}}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1} \operatorname{Cov}(\boldsymbol{b}) \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{a}} \\ & =\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{a}}+\boldsymbol{\Sigma}_{\boldsymbol{a}, \boldsymbol{b}} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{b}}^{-1} \boldsymbol{\Sigma}_{\boldsymbol{b}, \boldsymbol{a}}\\ &=\operatorname{Cov}(\boldsymbol{a} \mid \boldsymbol{b} =\boldsymbol{\beta}) \end{aligned} \]

Can we find a transformation that will turn a Gaussian *process* prior sample (i.e. function) into a Gaussian *process* posterior sample, and thus use prior samples, which are presumably pretty easy, to create posterior ones, which are often hard. If we evaluate the sampled function at a finite number of points, then we can use the Matheron formula to do precisely that. Sometimes this can even be useful. The resulting algorithm uses tricks from both analytic GP regression and Monte Carlo.

The sample based approximation to this is precisely the Ensemble Kalman Filter.

## 2 “Exact” updates for Gaussian processes

Exact in the sense that we do not approximate the *data*. These updates are not exact if our basis function representation is only an approximation to the “true” GP (as with classic GPs) and not exact in the sense that we will be using samples to approximate measures. For now we assume that the observation likelihood is Gaussian.^{1}

For a Gaussian process \(f \sim \mathcal{G P}(\mu, k)\) with marginal \(\boldsymbol{f}_{m}=f(\mathbf{Z})\), the process conditioned on \(\boldsymbol{f}_{m}=\boldsymbol{y}\) admits, in distribution, the representation \[ \underbrace{(f \mid \boldsymbol{y})(\cdot)}_{\text {posterior }} \stackrel{\mathrm{d}}{=} \underbrace{f(\cdot)}_{\text {prior }}+\underbrace{k(\cdot, \mathbf{Z}) \mathbf{K}_{m, m}^{-1}\left(\boldsymbol{y}-\boldsymbol{f}_{m}\right)}_{\text {update }}. \]

If our observations are contaminated by additive i.i.d Gaussian noise, \(\boldsymbol{y}=\boldsymbol{f}_{m} +\boldsymbol{\varepsilon}\) with \(\boldsymbol{\varepsilon}\sim\mathcal{N}(\boldsymbol{0}, \sigma^2\mathbf{I}),\) we find \[ \begin{aligned} &\boldsymbol{f}_{*} \mid \boldsymbol{y} \stackrel{\mathrm{d}}{=} \boldsymbol{f}_{*}+\mathbf{K}_{*, n}\left(\mathbf{K}_{n, n}+\sigma^{2} \mathbf{I}\right)^{-1}(\boldsymbol{y}-\boldsymbol{f}-\boldsymbol{\varepsilon}) \end{aligned} \] When sampling from exact GPs we jointly draw \(\boldsymbol{f}_{*}\) and \(\boldsymbol{f}\) from the prior. Then, we combine \(\boldsymbol{f}\) with noise variates \(\varepsilon \sim \mathcal{N}\left(\mathbf{0}, \sigma^{2} \mathbf{I}\right)\) such that \(\boldsymbol{f}+\varepsilon\) constitutes a draw from the prior distribution of \(\boldsymbol{y}\).

Compare this to the equivalent distributional update from classical GP regression which updates the *moments* of a *distribution*, not *samples from a path* — the formulae are related though:

…the conditional distribution is the Gaussian \(\mathcal{N}\left(\boldsymbol{\mu}_{* \mid y}, \mathbf{K}_{*, * \mid y}\right)\) with moments \[\begin{aligned} \boldsymbol{\mu}_{* \mid \boldsymbol{y}}&=\boldsymbol{\mu}_*+\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1}\left(\boldsymbol{y}-\boldsymbol{\mu}_n\right) \\ \mathbf{K}_{*, * \mid \boldsymbol{y}}&=\mathbf{K}_{*, *}-\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1} \mathbf{K}_{n, *}\end{aligned} \]

## 3 Using basis functions

For many purposes we need a basis function representation, a.k.a. the *weight-space* representation. We assert the GP can be written as a random function comprising basis functions \(\boldsymbol{\phi}=\left(\phi_{1}, \ldots, \phi_{\ell}\right)\) with the Gaussian random weight vector \(w\) so that \[
f^{(w)}(\cdot)=\sum_{i=1}^{\ell} w_{i} \phi_{i}(\cdot) \quad \boldsymbol{w} \sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Sigma}_{\boldsymbol{w}}\right).
\] \(f^{(w)}\) is a random function satisfying \(\boldsymbol{f}^{(\boldsymbol{w})} \sim \mathcal{N}\left(\mathbf{0}, \boldsymbol{\Phi}_{n} \boldsymbol{\Sigma}_{\boldsymbol{w}} \boldsymbol{\Phi}^{\top}\right)\), where \(\boldsymbol{\Phi}_{n}=\boldsymbol{\phi}(\mathbf{X})\) is a \(|\mathbf{X}| \times \ell\) matrix of features. If we are lucky, the representation might not be too bad when the basis is truncated to a small size.

The posterior weight distribution \(\boldsymbol{w} \mid \boldsymbol{y} \sim \mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{w} \mid n}, \boldsymbol{\Sigma}_{\boldsymbol{w} \mid n}\right)\) is Gaussian with moments \[ \begin{aligned} \boldsymbol{\mu}_{\boldsymbol{w} \mid n} &=\left(\boldsymbol{\Phi}^{\top} \boldsymbol{\Phi}+\sigma^{2} \mathbf{I}\right)^{-1} \boldsymbol{\Phi}^{\top} \boldsymbol{y} \\ \boldsymbol{\Sigma}_{\boldsymbol{w} \mid n} &=\left(\boldsymbol{\Phi}^{\top} \boldsymbol{\Phi}+\sigma^{2} \mathbf{I}\right)^{-1} \sigma^{2} \end{aligned} \] where \(\boldsymbol{\Phi}=\boldsymbol{\phi}(\mathbf{X})\) is an \(n \times \ell\) feature matrix. We solve for the right-hand side at \(\mathcal{O}\left(\min \{\ell, n\}^{3}\right)\) cost by applying the Woodbury identity as needed. So far there is nothing unusual here; the cool bit is realising we can represent this update as an independent operation.

In the weight-space setting, the pathwise update given an initial weight vector \(\boldsymbol{w} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})\) is \(\boldsymbol{w} \mid \boldsymbol{y} \stackrel{\mathrm{d}}{=} \boldsymbol{w}+\boldsymbol{\Phi}^{\top}\left(\boldsymbol{\Phi} \boldsymbol{\Phi}^{\top}+\sigma^{2} \mathbf{I}\right)^{-1}\left(\boldsymbol{y}-\boldsymbol{\Phi}^{\top} \boldsymbol{w}-\boldsymbol{\varepsilon}\right).\)

So if we had a nice weight-space representation for everything already we could go home at this point. However, for many models we are not given that; we might find natural bases for the prior and posterior are not the same (the posterior should not be stationary usually, for one thing).

The innovation in J. T. Wilson et al. (2020) is to make different choices of functional bases for prior and posterior updates. We can choose anything really, AFAICT. They suggest Fourier basis for prior and the *canonical basis*, i.e. the reproducing kernel basis \(k(\cdot,\vv{x})\) for the update. Then we have \[
\underbrace{(f \mid \boldsymbol{y})(\cdot)}_{\text {sparse posterior }} \stackrel{\mathrm{d}}{\approx} \underbrace{\sum_{i=1}^{\ell} w_{i} \phi_{i}(\cdot)}_{\text {weight-space prior}} +\underbrace{\sum_{j=1}^{m} v_{j} k\left(\cdot, \boldsymbol{x}_{j}\right)}_{\text {function-space update}} ,
\] where we have defined \(\boldsymbol{v}=\left(\mathbf{K}_{n, n}+\sigma^{2} \mathbf{I}\right)^{-1}\left(\boldsymbol{y}-\boldsymbol{\Phi}^{\top} \boldsymbol{w}- \boldsymbol{\varepsilon}\right) .\)

## 4 Sparse GPs

i.e. using inducing variables.

Given \(q(\boldsymbol{u})\), we approximate posterior distributions as \[ p\left(\boldsymbol{f}_{*} \mid \boldsymbol{y}\right) \approx \int_{\mathbb{R}^{m}} p\left(\boldsymbol{f}_{*} \mid \boldsymbol{u}\right) q(\boldsymbol{u}) \mathrm{d} \boldsymbol{u} . \] If \(\boldsymbol{u} \sim \mathcal{N}\left(\boldsymbol{\mu}_{\boldsymbol{u}}, \boldsymbol{\Sigma}_{\boldsymbol{u}}\right)\), we compute this integral analytically to obtain a Gaussian distribution with mean and covariance \[ \begin{aligned} \boldsymbol{m}_{* \mid m} &=\mathbf{K}_{*, m} \mathbf{K}_{m, m}^{-1} \boldsymbol{\mu}_{m} \\ \mathbf{K}_{*, * \mid m} &=\mathbf{K}_{*, *}+\mathbf{K}_{*, m} \mathbf{K}_{m, m}^{-1}\left(\boldsymbol{\Sigma}_{\boldsymbol{u}}-\mathbf{K}_{m, m}\right) \mathbf{K}_{m, m}^{-1} \mathbf{K}_{m, *^{*}} \end{aligned} \]

\[ \begin{aligned} &\boldsymbol{f}_{*} \mid \boldsymbol{u} \stackrel{\mathrm{d}}{=} \boldsymbol{f}_{*}+\mathbf{K}_{*, m} \mathbf{K}_{m, m}^{-1}\left(\boldsymbol{u}-\boldsymbol{f}_{m}\right) \\ \end{aligned} \]

When sampling from sparse GPs we draw \(\boldsymbol{f}_{*}\) and \(\boldsymbol{f}_{m}\) together from the prior, and independently generate target values \(\boldsymbol{u} \sim q(\boldsymbol{u}) .\) \[ \underbrace{(f \mid \boldsymbol{u})(\cdot)}_{\text {sparse posterior }} \stackrel{\mathrm{d}}{\approx} \underbrace{\sum_{i=1}^{\ell} w_{i} \phi_{i}(\cdot)}_{\text {weight-space prior}} +\underbrace{\sum_{j=1}^{m} v_{j} k\left(\cdot, \boldsymbol{z}_{j}\right)}_{\text {function-space update}} , \] where we have defined \(\boldsymbol{v}=\mathbf{K}_{m, m}^{-1}\left(\boldsymbol{u}-\boldsymbol{\Phi}^{\top} \boldsymbol{w}\right).\)

## 5 More pathwise hacks

Conditioning is all well and good, but what else might we want to do with pathwise updates? One possible use is *density multiplication*: Given \(\boldsymbol{a}_1\sim \mathcal{N}(\boldsymbol{\mu}_1,\boldsymbol{\Sigma}_1)\) and \(\boldsymbol{a}_2\sim \mathcal{N}(\boldsymbol{\mu}_2,\boldsymbol{\Sigma}_2)\), can I find a pathwise update that gives me \(\boldsymbol{a}_{1,2}\sim \mathcal{N}(\boldsymbol{\mu}_1,\boldsymbol{\Sigma}_1)\mathcal{N}(\boldsymbol{\mu}_2,\boldsymbol{\Sigma}_2)=:\mathcal{N}(\boldsymbol{\mu}_{12},\boldsymbol{\Sigma}_{12})\)? This ends up being useful in, e.g. belief propagation. A standard result is that the product of two Gaussian densities is another Gaussian density, with mean and covariance given by $

To keep this simple let’s consider the case where \(\boldsymbol{\mu}_1=\boldsymbol{\mu}_1=0\).

## 6 Noisy observations

TBD

## 7 What pathwise updates are possible?

Another way we can think about this is in terms of the action on \(\boldsymbol{x}\), and the correlated \(\boldsymbol{y}\) as an auxiliary which we use to update \(\boldsymbol{x}\). Does that get us anything useful?

## 8 Matrix GPs

(Ritter et al. 2021 appendix D) reframes the Matheron update and generalises it to matrix Gaussians. TBC.

## 9 Stationary moves

Thus far we have talked about moves updating a prior to a posterior; how about moves *within* a posterior?

We could try Langevin sampling, for example, or SG MCMC but these all seem to require inverting the covariance matrix so are not likely to be efficient in general. Can we do better?

## 10 Incoming

- Alexander Terenin, Pathwise Conditioning and Non-Euclidean Gaussian Processes

## 11 References

*Canadian Journal of Statistics*.

*Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence*. UAI ’04.

*arXiv:1705.07104 [Cs, Stat]*.

*arXiv:1403.6015 [Astro-Ph, Stat]*.

*IEEE Transactions on Information Theory*.

*arXiv:1805.00753 [Stat]*.

*Probability Theory and Related Fields*.

*Proceedings of the 20th International Conference on Neural Information Processing Systems*. NIPS’07.

*Journal of Machine Learning Research*.

*arXiv:2006.10160 [Cs, Stat]*.

*Journal of Machine Learning Research*.

*2016 International Joint Conference on Neural Networks (IJCNN)*.

*Mathematical Geology*.

*Statistics for Spatial Data*.

*Statistics for Spatio-Temporal Data*. Wiley Series in Probability and Statistics 2.0.

*Neural Computation*.

*Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic*. NIPS’01.

*Proceedings of the 25th International Conference on Machine Learning*. ICML ’08.

*PMLR*.

*Data Analytics for Renewable Energy Integration: Informing the Generation and Distribution of Renewable Energy*. Lecture Notes in Computer Science.

*arXiv:1903.03986 [Cs, Stat]*.

*Artificial Intelligence and Statistics*.

*Advances in Neural Information Processing Systems 24*.

*Advances in Neural Information Processing Systems 28*. NIPS’15.

*arXiv:2012.00152 [Cs, Stat]*.

*Handbook of Mathematical Geosciences: Fifty Years of IAMG*.

*Journal of Machine Learning Research*.

*arXiv:2105.04504 [Cs, Stat]*.

*Proceedings of the 30th International Conference on Machine Learning (ICML-13)*.

*arXiv:1505.02965 [Math, Stat]*.

*Advances in Neural Information Processing Systems 30*.

*Mathematical Geology*.

*Journal of Machine Learning Research*.

*The Annals of Statistics*.

*arXiv:2010.10876 [Cs]*.

*arXiv:1711.00799 [Stat]*.

*Advances in Neural Information Processing Systems 27*.

*Advances in Neural Information Processing Systems 26*.

*Proceedings of the 33rd International Conference on Machine Learning (ICML-16)*.

*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*.

*arXiv:1402.1412 [Stat]*.

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*. NIPS’18.

*arXiv:1802.08903 [Cs, Stat]*.

*arXiv:1807.01613 [Cs, Stat]*.

*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*.

*Proceedings of the 22nd International Conference on Machine Learning - ICML ’05*.

*Journal of Statistical Software*.

*Journal of Computational and Graphical Statistics*.

*Handbook of Uncertainty Quantification*.

*Proceedings of the Conference on Uncertainty in Artificial Intelligence*.

*2010 IEEE International Workshop on Machine Learning for Signal Processing*.

*Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence*. UAI’13.

*Pattern Recognition Letters*.

*arXiv:1806.10234 [Cs, Stat]*.

*Conference on Uncertainty in Artificial Intelligence*.

*Learning in Graphical Models*.

*2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP)*.

*arXiv:2001.08055 [Physics, Stat]*.

*ICLR 2014 Conference*.

*Mathematical and Computer Modelling of Dynamical Systems*.

*Autonomous Robots*.

*UAI17*.

*Journal of the Southern African Institute of Mining and Metallurgy*.

*arXiv:1308.0399 [Stat]*.

*Journal of Machine Learning Research*.

*Proceedings of the 16th Annual Conference on Neural Information Processing Systems*.

*Proceedings of the 26th Annual International Conference on Machine Learning*. ICML ’09.

*Journal of Machine Learning Research*.

*ICLR*.

*Journal of Computational and Graphical Statistics*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*IEEE Transactions on Signal Processing*.

*Twenty-Eighth AAAI Conference on Artificial Intelligence*.

*Advances in Neural Information Processing Systems*.

*NATO ASI Series. Series F: Computer and System Sciences*.

*Information Theory, Inference & Learning Algorithms*.

*Traité de Géostatistique Appliquée. 2. Le Krigeage*.

*Economic Geology*.

*arXiv:1610.08733 [Stat]*.

*Proceedings of ICLR*.

*Journal of Process Control*, DYCOPS-CAB 2016,.

*Journal of Machine Learning Research*.

*Neural Computation*.

*SIAM/ASA Journal on Uncertainty Quantification*.

*arXiv:2104.14987 [Stat]*.

*arXiv:1911.00002 [Cs, Stat]*.

*SSRN Electronic Journal*.

*International Conference on Machine Learning*.

*Journal of the Royal Statistical Society: Series B (Methodological)*.

*Biometrika*.

*Journal of Open Source Software*.

*Advances in Neural Information Processing Systems*.

*Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence*. UAI’10.

*Journal of Machine Learning Research*.

*arXiv:1701.02440 [Cs, Math, Stat]*.

*Gaussian Processes for Machine Learning*. Adaptive Computation and Machine Learning.

*2010 13th International Conference on Information Fusion*.

*arXiv:2105.14594 [Cs, Stat]*.

*arXiv:2004.11408 [Stat]*.

*Proceedings of The 24th International Conference on Artificial Intelligence and Statistics*.

*Proceedings of the 27th International Conference on International Conference on Machine Learning*. ICML’10.

*arXiv:1910.09349 [Cs, Stat]*.

*Advances In Neural Information Processing Systems*.

*International Conference on Artificial Intelligence and Statistics*.

*Artificial Neural Networks and Machine Learning – ICANN 2011*. Lecture Notes in Computer Science.

*Bayesian Filtering and Smoothing*. Institute of Mathematical Statistics Textbooks 3.

*Artificial Intelligence and Statistics*.

*IEEE Signal Processing Magazine*.

*Proceedings of the 31st International Conference on Neural Information Processing Systems*. NIPS’17.

*Artificial Intelligence and Statistics*.

*Scalable Bayesian Spatial Analysis with Gaussian Markov Random Fields*. Linköping Studies in Statistics.

*arXiv:1809.02010 [Cs, Stat]*.

*Advances in Neural Information Processing Systems*.

*Statistics and Computing*.

*arXiv:2006.15641 [Cs, Stat]*.

*arXiv:1908.05726 [Math, Stat]*.

*International Conference on Artificial Intelligence and Statistics*.

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*.

*Journal of Computational and Graphical Statistics*.

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*.

*IEEE Transactions on Signal Processing*.

*NIPS 2014 Workshop on Advances in Variational Inference*.

*Journal of Machine Learning Research*.

*arXiv:1206.5754 [Cs, Stat]*.

*Proceedings of the 25th International Conference on Machine Learning*. ICML ’08.

*Computer Graphics Forum*.

*Advances in Neural Information Processing Systems*.

*Spatio-Temporal Statistics with R*.

*arXiv:1901.11436 [Cs, Eess, Stat]*.

*Advances in Neural Information Processing Systems 21*.

*Advances in Neural Information Processing Systems*.

*International Conference on Machine Learning*.

*Proceedings of the 37th International Conference on Machine Learning*.

*Journal of Machine Learning Research*.

*arXiv:1510.07389 [Cs, Stat]*.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*. UAI’11.

*Machine Learning and Knowledge Discovery in Databases*. Lecture Notes in Computer Science.

*Proceedings of the 29th International Coference on International Conference on Machine Learning*. ICML’12.

*Proceedings of the 32Nd International Conference on International Conference on Machine Learning - Volume 37*. ICML’15.

*Proceedings of NeurIPS 2020*.

## Footnotes

There are neat extensions to the non-Gaussian and sparse cases; that comes later.↩︎