- Quick intro
- Observation likelihoods
- Incorporating a mean function
- Density estimation
- Kernels
- Using state filtering
- On lattice observations
- On manifolds
- By variational inference
- With inducing variables
- By variational inference with inducing variables
- With vector output
- Deep
- Approximation with dropout
- Inhomogeneous with covariates
- For dimension reduction
- Pathwise/Matheron updates
- Implementations
- References

Gaussian random processes/fields are stochastic processes/fields with jointly Gaussian distributions of observations.
While “Gaussian *process* regression” is not wrong *per se*, there is a common convention in stochastic process theory (and also in pedagogy) to use *process* to talk about some notionally time-indexed process and *field* to talk about ones that have a some space-like index without a presumption of an arrow of time.
This leads to much confusion, because Gaussian *field* regression is what we usually want to talk about. What we want to use the arrow of time for is a whole other story.
Regardless, hereafter I’ll use “field” and “process” interchangeably.

In machine learning, Gaussian fields are used often as a means of regression or classification, since it is fairly easy to conditionalize a Gaussian field on data and produce a posterior distribution over functions. They provide nonparametric method of inferring regression functions, with a conveniently Bayesian interpretation and reasonably elegant learning and inference steps. I would further add that this is the crystal meth of machine learning methods, in terms of the addictiveness, and of the passion of the people who use it.

The central trick is using a clever union of Hilbert space trickss and probability to give a probabilistic interpretation of functional regression as a kind of nonparametric Bayesian inference.

Useful side divergence into
representer theorems and Karhunen-Loève expansions for thinking about this.
Regression using Gaussian processes is common
e.g. spatial statistics
where it arises as *kriging*.
Cressie (1990) traces a history of this idea via Matheron (1963a), to works of Krige (1951).

This web site aims to provide an overview of resources concerned with probabilistic modeling, inference and learning based on Gaussian processes. Although Gaussian processes have a long history in the field of statistics, they seem to have been employed extensively only in niche areas. With the advent of kernel machines in the machine learning community, models based on Gaussian processes have become commonplace for problems of regression (kriging) and classification as well as a host of more specialized applications.

I’ve not been enthusiastic about these in the past. It’s nice to have a principled nonparametric Bayesian formalism, but it has always seemed pointless having a formalism that is so computationally demanding that people don’t try to use more than a thousand data points, or spend most of a paper working out how to approximate this simple elegant model with a complex messy model. However, that previous sentence describes most of my career now, so I guess I must have come around.

Perhaps I should be persuaded by tricks such as AutoGP (Krauth et al. 2016) which breaks some computational deadlocks by clever use of inducing variables and variational approximation to produce a compressed representation of the data with tractable inference and model selection, including kernel selection, and doing the whole thing in many dimensions simultaneously. There are other clever tricks like this one, e.g (Saatçi 2012) shows how to use a lattice structure for observations to make computation cheap.

## Quick intro

I am not the right guy to provide the canonical introduction, because it already exists. Specifically, Rasmussen and Williams (2006).

This lecture by the late David Mackay is probably good; the man could talk.

There is also a well-illustrated and elementary introduction by Yuge Shi. There are many, many more.

J. T. Wilson et al. (2021):

A Gaussian process (GP) is a random function \(f: \mathcal{X} \rightarrow \mathbb{R}\), such that, for any finite collection of points \(\mathbf{X} \subset \mathcal{X}\), the random vector \(\boldsymbol{f}=f(\mathbf{X})\) follows a Gaussian distribution. Such a process is uniquely identified by a mean function \(\mu: \mathcal{X} \rightarrow \mathbb{R}\) and a positive semi-definite kernel \(k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}\). Hence, if \(f \sim \mathcal{G} \mathcal{P}(\mu, k)\), then \(\boldsymbol{f} \sim \mathcal{N}(\boldsymbol{\mu}, \mathbf{K})\) is multivariate normal with mean \(\boldsymbol{\mu}=\mu(\mathbf{X})\) and covariance \(\mathbf{K}=k(\mathbf{X}, \mathbf{X})\).

[…] we investigate different ways of reasoning about the random variable \(\boldsymbol{f}_* \mid \boldsymbol{f}_n=\boldsymbol{y}\) for some non-trivial partition \(\boldsymbol{f}=\boldsymbol{f}_n \oplus \boldsymbol{f}_*\). Here, \(\boldsymbol{f}_n=f\left(\mathbf{X}_n\right)\) are process values at a set of training locations \(\mathbf{X}_n \subset \mathbf{X}\) where we would like to introduce a condition \(\boldsymbol{f}_n=\boldsymbol{y}\), while \(\boldsymbol{f}_*=f\left(\mathbf{X}_*\right)\) are process values at a set of test locations \(\mathbf{X}_* \subset \mathbf{X}\) where we would like to obtain a random variable \(\boldsymbol{f}_* \mid \boldsymbol{f}_n=\boldsymbol{y}\).

[…] we may obtain \(\boldsymbol{f}_* \mid \boldsymbol{y}\) by first finding its conditional distribution. Since process values \(\left(\boldsymbol{f}_n, \boldsymbol{f}_*\right)\) are defined as jointly Gaussian, this procedure closely resembles that of [the finite-dimensinal case]: we factor out the marginal distribution of \(\boldsymbol{f}_n\) from the joint distribution \(p\left(\boldsymbol{f}_n, \boldsymbol{f}_*\right)\) and, upon canceling, identify the remaining distribution as \(p\left(\boldsymbol{f}_* \mid \boldsymbol{y}\right)\). Having done so, we find that the conditional distribution is the Gaussian \(\mathcal{N}\left(\boldsymbol{\mu}_{* \mid y}, \mathbf{K}_{*, * \mid y}\right)\) with moments \[\begin{aligned} \boldsymbol{\mu}_{* \mid \boldsymbol{y}}&=\boldsymbol{\mu}_*+\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1}\left(\boldsymbol{y}-\boldsymbol{\mu}_n\right) \\ \mathbf{K}_{*, * \mid \boldsymbol{y}}&=\mathbf{K}_{*, *}-\mathbf{K}_{*, n} \mathbf{K}_{n, n}^{-1} \mathbf{K}_{n, *}\end{aligned} \]

## Observation likelihoods

classification etc

## Incorporating a mean function

Almost immediate but not *quite* trivial
(Rasmussen and Williams 2006, 2.7).

TODO: discuss identifiability.

## Density estimation

Can I infer a density using GPs? Yes. One popular method is apparently the logistic Gaussian process. (Tokdar 2007; Lenk 2003)

## Kernels

a.k.a. covariance models.

GP regression models are kernel machines. As such covariance kernels are the parameters. More or less. One can also parameterise with a mean function, but let us ignore that detail for now because usually we do not use them.

## Using state filtering

When one dimension of the input vector can be interpreted as a time dimension we are Kalman filtering Gaussian Processes, which has benefits in terms of speed.

## On lattice observations

## On manifolds

I would like to read Terenin on GPs on Manifolds who also makes a suggestive connection to SDEs, which is the filtering GPs trick again.

## By variational inference

🏗

## With inducing variables

“Sparse GP”.
See (**Quinonero-CandelaUnifying2005?**).
🏗

## By variational inference with inducing variables

See GP factoring.

## With vector output

## Approximation with dropout

See NN ensembles.

## Inhomogeneous with covariates

Integrated nested Laplace approximation connects to GP-as-SDE idea, I think?

## For dimension reduction

e.g. GP-LVM (N. Lawrence 2005). 🏗

## Pathwise/Matheron updates

See pathwise GP.

## Implementations

## 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]*, May.

*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 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*.

*arXiv:1906.08324 [Cs, Stat]*, June.

*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.

*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 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*.

## No comments yet. Why not leave one?