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

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.

**Danger**: notation updates in the pipeline.

## 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.

## “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 \[\begin{aligned} \boldsymbol{\mu}_{* \mid \boldsymbol{y}}&=\boldsymbol{\mu}_*+\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1}\left(\boldsymbol{y}-\boldsymbol{\mu}_n\right) \\ \mathbf{K}_{*, * \mid \boldsymbol{y}}&=\mathbf{K}_{*, *}-\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1} \mathbf{K}_{n, *}\end{aligned} \]

## 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 \(\boldsymbol{v}=\left(\mathbf{K}_{n, n}+\sigma^{2} \mathbf{I}\right)^{-1}\left(\boldsymbol{y}-\boldsymbol{\Phi}^{\top} \boldsymbol{w}- \boldsymbol{\varepsilon}\right) .\)

## Sparse GP setting

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 \(\boldsymbol{v}=\mathbf{K}_{m, m}^{-1}\left(\boldsymbol{u}-\boldsymbol{\Phi}^{\top} \boldsymbol{w}\right) .\)

## Matrix GPs

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

## 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?

## Incoming

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

## References

*Canadian Journal of Statistics*26 (1): 127–37.

*Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence*, 2–9. UAI ’04. Arlington, Virginia, United States: AUAI Press.

*arXiv:1705.07104 [Cs, Stat]*, November.

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

*IEEE Transactions on Information Theory*64 (10): 6620–37.

*arXiv:1805.00753 [Stat]*, April.

*Probability Theory and Related Fields*138 (1-2): 33–73.

*Proceedings of the 20th International Conference on Neural Information Processing Systems*, 153–60. NIPS’07. USA: Curran Associates Inc.

*Journal of Machine Learning Research*20 (117): 1–63.

*arXiv:2006.10160 [Cs, Stat]*, June.

*Journal of Machine Learning Research*21 (131): 1–63.

*2016 International Joint Conference on Neural Networks (IJCNN)*, 3338–45. Vancouver, BC, Canada: IEEE.

*Mathematical Geology*22 (3): 239–52.

*Statistics for Spatial Data*. John Wiley & Sons.

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

*Neural Computation*14 (3): 641–68.

*Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic*, 657–63. NIPS’01. Cambridge, MA, USA: MIT Press.

*Proceedings of the 25th International Conference on Machine Learning*, 192–99. ICML ’08. New York, NY, USA: ACM Press.

*PMLR*.

*Data Analytics for Renewable Energy Integration: Informing the Generation and Distribution of Renewable Energy*, edited by Wei Lee Woon, Zeyar Aung, Oliver Kramer, and Stuart Madnick, 94–106. Lecture Notes in Computer Science. Cham: Springer International Publishing.

*arXiv:1903.03986 [Cs, Stat]*, March.

*Artificial Intelligence and Statistics*, 207–15.

*Advances in Neural Information Processing Systems 24*, edited by J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger, 2510–18. Curran Associates, Inc.

*Advances in Neural Information Processing Systems 28*, 1414–22. NIPS’15. Cambridge, MA, USA: MIT Press.

*arXiv:2012.00152 [Cs, Stat]*, November.

*Handbook of Mathematical Geosciences: Fifty Years of IAMG*, edited by B.S. Daya Sagar, Qiuming Cheng, and Frits Agterberg, 3–24. Cham: Springer International Publishing.

*Journal of Machine Learning Research*19 (1): 2100–2145.

*arXiv:2105.04504 [Cs, Stat]*.

*Proceedings of the 30th International Conference on Machine Learning (ICML-13)*, 1166–74.

*arXiv:1505.02965 [Math, Stat]*, May.

*Advances in Neural Information Processing Systems 30*, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5309–19. Curran Associates, Inc.

*Mathematical Geology*39 (6): 607–23.

*Journal of Machine Learning Research*6 (Apr): 615–37.

*The Annals of Statistics*1 (2): 209–30.

*arXiv:2010.10876 [Cs]*, October.

*arXiv:1711.00799 [Stat]*, November.

*Advances in Neural Information Processing Systems 27*, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 3680–88. Curran Associates, Inc.

*Advances in Neural Information Processing Systems 26*, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, 3156–64. Curran Associates, Inc.

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

*arXiv:1402.1412 [Stat]*, February.

*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*, 353–61. PMLR.

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*, 31:7587–97. NIPS’18. Red Hook, NY, USA: Curran Associates Inc.

*arXiv:1802.08903 [Cs, Stat]*, February.

*arXiv:1807.01613 [Cs, Stat]*, July, 10.

*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*371 (1984): 20110553.

*IEEE Transactions on Pattern Analysis and Machine Intelligence*37 (2): 424–36.

*Proceedings of the 22nd International Conference on Machine Learning - ICML ’05*, 241–48. Bonn, Germany: ACM Press.

*Journal of Statistical Software*72 (1).

*Journal of Computational and Graphical Statistics*24 (2): 561–78.

*Handbook of Uncertainty Quantification*, edited by Roger Ghanem, David Higdon, and Houman Owhadi, 1–37. Cham: Springer International Publishing.

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

*2010 IEEE International Workshop on Machine Learning for Signal Processing*, 379–84. Kittila, Finland: IEEE.

*Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence*, 282–90. UAI’13. Arlington, Virginia, USA: AUAI Press.

*Pattern Recognition Letters*45 (August): 85–91.

*arXiv:1806.10234 [Cs, Stat]*, June.

*Conference on Uncertainty in Artificial Intelligence*, 789–98. PMLR.

*Learning in Graphical Models*. Cambridge, Mass.: MIT Press.

*2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP)*, 1–6. Vietri sul Mare, Salerno, Italy: IEEE.

*arXiv:2001.08055 [Physics, Stat]*, January.

*ICLR 2014 Conference*.

*Autonomous Robots*, 27:75–90.

*Mathematical and Computer Modelling of Dynamical Systems*11 (4): 411–24.

*UAI17*.

*Journal of the Southern African Institute of Mining and Metallurgy*52 (6): 119–39.

*arXiv:1308.0399 [Stat]*, August.

*Journal of Machine Learning Research*6 (Nov): 1783–1816.

*Proceedings of the 26th Annual International Conference on Machine Learning*, 601–8. ICML ’09. New York, NY, USA: ACM.

*Proceedings of the 16th Annual Conference on Neural Information Processing Systems*, 609–16.

*Journal of Machine Learning Research*11 (Jun): 1865–81.

*ICLR*.

*Journal of Computational and Graphical Statistics*12 (3): 548–65.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*73 (4): 423–98.

*IEEE Transactions on Signal Processing*59 (7): 3155–67.

*Twenty-Eighth AAAI Conference on Artificial Intelligence*.

*Advances in Neural Information Processing Systems*. Vol. 32. Curran Associates, Inc.

*NATO ASI Series. Series F: Computer and System Sciences*168: 133–65.

*Information Theory, Inference & Learning Algorithms*, Chapter 45. Cambridge University Press.

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

*Economic Geology*58 (8): 1246–66.

*arXiv:1610.08733 [Stat]*, October.

*Journal of Process Control*, DYCOPS-CAB 2016, 60 (December): 82–94.

*Proceedings of ICLR*.

*Journal of Machine Learning Research*6 (Jul): 1099–1125.

*Neural Computation*17 (1): 177–204.

*SIAM/ASA Journal on Uncertainty Quantification*, February, 96–124.

*arXiv:2104.14987 [Stat]*, April.

*arXiv:1911.00002 [Cs, Stat]*, October.

*SSRN Electronic Journal*.

*International Conference on Machine Learning*, 3789–98.

*Journal of the Royal Statistical Society: Series B (Methodological)*40 (1): 1–24.

*Biometrika*99 (3): 511–31.

*Journal of Open Source Software*7 (75): 4455.

*Advances in Neural Information Processing Systems*33.

*Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence*, 450–57. UAI’10. Arlington, Virginia, USA: AUAI Press.

*Journal of Machine Learning Research*6 (Dec): 1939–59.

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

*Gaussian Processes for Machine Learning*. Adaptive Computation and Machine Learning. Cambridge, Mass: MIT Press.

*2010 13th International Conference on Information Fusion*, 1–9.

*arXiv:2105.14594 [Cs, Stat]*, May.

*arXiv:2004.11408 [Stat]*, April.

*Proceedings of The 24th International Conference on Artificial Intelligence and Statistics*, 1837–45. PMLR.

*Proceedings of the 27th International Conference on International Conference on Machine Learning*, 927–34. ICML’10. Madison, WI, USA: Omnipress.

*arXiv:1910.09349 [Cs, Stat]*, March.

*Advances In Neural Information Processing Systems*.

*International Conference on Artificial Intelligence and Statistics*, 689–97.

*Artificial Neural Networks and Machine Learning – ICANN 2011*, edited by Timo Honkela, Włodzisław Duch, Mark Girolami, and Samuel Kaski, 6792:151–58. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer.

*Bayesian Filtering and Smoothing*. Institute of Mathematical Statistics Textbooks 3. Cambridge, U.K. ; New York: Cambridge University Press.

*Artificial Intelligence and Statistics*.

*IEEE Signal Processing Magazine*30 (4): 51–61.

*Proceedings of the 31st International Conference on Neural Information Processing Systems*, 1696–706. NIPS’17. Red Hook, NY, USA: Curran Associates Inc.

*Artificial Intelligence and Statistics*, 877–85. PMLR.

*Scalable Bayesian Spatial Analysis with Gaussian Markov Random Fields*. Vol. 15. Linköping Studies in Statistics. Linköping: Linköping University Electronic Press.

*arXiv:1809.02010 [Cs, Stat]*, September.

*Advances in Neural Information Processing Systems*, 1257–64.

*Statistics and Computing*30 (2): 419–46.

*arXiv:2006.15641 [Cs, Stat]*, June.

*arXiv:1908.05726 [Math, Stat]*, August.

*International Conference on Artificial Intelligence and Statistics*, 567–74. PMLR.

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*, 844–51.

*Journal of Computational and Graphical Statistics*16 (3): 633–55.

*IEEE Transactions on Signal Processing*62 (23): 6171–83.

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*, 868–75.

*Journal of Machine Learning Research*14 (April): 1175−1179.

*arXiv:1206.5754 [Cs, Stat]*, July.

*Proceedings of the 25th International Conference on Machine Learning*, 1112–19. ICML ’08. New York, NY, USA: ACM.

*Computer Graphics Forum*25 (3): 635–44.

*Advances in Neural Information Processing Systems*, 32:14648–59. Red Hook, NY, USA.

*Spatio-Temporal Statistics with R*.

*NIPS 2014 Workshop on Advances in Variational Inference*.

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

*Advances in Neural Information Processing Systems*, 682–88.

*Advances in Neural Information Processing Systems 21*, edited by D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, 265–72. Curran Associates, Inc.

*International Conference on Machine Learning*.

*arXiv:1510.07389 [Cs, Stat]*, October.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press.

*Machine Learning and Knowledge Discovery in Databases*, edited by Peter A. Flach, Tijl De Bie, and Nello Cristianini, 858–61. Lecture Notes in Computer Science. Springer Berlin Heidelberg.

*Proceedings of the 29th International Coference on International Conference on Machine Learning*, 1139–46. ICML’12. Madison, WI, USA: Omnipress.

*Proceedings of the 32Nd International Conference on International Conference on Machine Learning - Volume 37*, 1775–84. ICML’15. Lille, France: JMLR.org.

*Proceedings of the 37th International Conference on Machine Learning*, 10292–302. PMLR.

*Journal of Machine Learning Research*22 (105): 1–47.

*Proceedings of NeurIPS 2020*.

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

## No comments yet. Why not leave one?