The Living Thing / Notebooks :

Gaussian process regression and classification

“Gaussian processes” are processes with Gaussian conditional distributions, like Brownian motions and suchlike. Very prominent in, e.g. spatial statistics where they are used for kriging etc.

However, when you see it capitalised it seems to means some specific emphasis, on the use of these processes for regression, as nonparametric method with a conveniently Bayesian interpretation?

I feel this is not too complex but I’ve never looked in to it. They reputedly work well with kernel methods to do machine learning stuff, apparently. The details of this are still hazy to me, and they aren’t currently on the correct side of the hype curve for me to dive in.

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.

The current scikit-learn has fancy gaussian processes, and an introduction.

Gaussian Processes (GP) are a generic supervised learning method designed to solve regression and probabilistic classification problems.

The advantages of Gaussian processes are:

The disadvantages of Gaussian processes include:


Covariance models

Covariance estimation is weird. The Matérn stationary (and in the Euclidean case, isotropic) covariance function is one model for covariance. See Carl Edward Rasmussen’s Gaussian Process lecture notes for a readable explanation, or chapter 4 of his textbook (RaWi06).

Connection to Kalman filtering

Looks interesting. Without knowing enough about either to make an informed judgement, I imagine this makes the Gaussian process regression soluble by making it local, i.e. Markov, with respect to some assumed hidden state, in the same way Kalman filtering does Wiener filtering. This would address at least some of the criticisms about sparsity etc.

See Simo Särkkä’s work for that. (HaSä10, SäHa12,SäSH13_, KaSä16)


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


Abrahamsen, P. (1997) A review of Gaussian random fields and correlation functions.
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Cutajar, K., Bonilla, E. V., Michiardi, P., & Filippone, M. (2016) Practical Learning of Deep Gaussian Processes via Random Fourier Features. arXiv:1610.04386 [Stat].
Duvenaud, D., Lloyd, J., Grosse, R., Tenenbaum, J., & Zoubin, G. (2013) Structure Discovery in Nonparametric Regression through Compositional Kernel Search. In Proceedings of the 30th International Conference on Machine Learning (ICML-13) (pp. 1166–1174).
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Grosse, R., Salakhutdinov, R. R., Freeman, W. T., & Tenenbaum, J. B.(2012) Exploiting compositionality to explore a large space of model structures. In Proceedings of the Conference on Uncertainty in Artificial Intelligence.
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Matthews, A. G. de G., van der Wilk, M., Nickson, T., Fujii, K., Boukouvalas, A., León-Villagrá, P., … Hensman, J. (2016) GPflow: A Gaussian process library using TensorFlow. arXiv:1610.08733 [Stat].
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Särkkä, S. (2013) Bayesian filtering and smoothing. . Cambridge, U.K.; New York: Cambridge University Press
Särkkä, S., & Hartikainen, J. (2012) Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression. In Journal of Machine Learning Research.
Särkkä, S., Solin, A., & Hartikainen, J. (2013) Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering. IEEE Signal Processing Magazine, 30(4), 51–61. DOI.
Snelson, E., & Ghahramani, Z. (2005) Sparse Gaussian processes using pseudo-inputs. In Advances in neural information processing systems (pp. 1257–1264).
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