# Gaussian process regression

### And possibly classification or other learning approaches

Usefulness: 🔧 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

Chi Feng’s amazing GP demo.

“Gaussian Processes” are stochastic processes/fields with jointly Gaussian distributions of observations. When you see it capitalised it tends to means a specific emphasis, on the use of these processes for regression, as nonparametric method with a conveniently Bayesian interpretation. The basic trick is using a clever union of Hilbert spaces and probability to give a probabilistic interpretation of functional regression as a kind of nonparametric Bayesian posterior inference, where one gets distributions over posterior functions. Regression using Gaussian processes is common e.g. spatial statistics where it arises as kriging.

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 very enthusiastic about these in the past. It’s nice to have a principled nonparametric Bayesian formalism, but it’s pointless having a formalism that is so computationally demanding that people don’t try to use more than a thousand datapoints.

However, 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.

## Density estimation

Can I infer a density using these? Yes. One popular method is the logsitic gaussian process. (Tokdar 2007; Lenk 2003)

## Kernels

a.k.a. covariance models.

GP models are the meeting of Covariance estimation and kernel machines.

🚧

“Sparse GP”. 🚧

## Approximation with variational inference and inducing variables

This combination is what makes AutoGP work. (Krauth et al. 2016). 🚧

## Dimension reduction

e.g. GP-LVM (Lawrence 2005). 🚧

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.

## Implementations

Bayes workhorse Stan can do Gaussian Process regression just like everything else; see Michael Betancourt’s blog, 1. 2. 3.

The current scikit-learn has semi-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 prediction interpolates the observations (at least for regular kernels).

• The prediction is probabilistic (Gaussian) so that one can compute empirical confidence intervals and decide based on those if one should refit (online fitting, adaptive fitting) the prediction in some region of interest.

• Versatile: different kernels can be specified. Common kernels are provided, but it is also possible to specify custom kernels.

The disadvantages of Gaussian processes include:

• They are not sparse, i.e., they use the whole samples/features information to perform the prediction.

• They lose efficiency in high dimensional spaces – namely when the number of features exceeds a few dozens.

Those last couple of points are not strictly correct; GPs can be made, in various senses, sparse. Also the scaling costs due to dimensionality of the features is swamped by the scaling costs of the number of data points. This kind of halfarsery is worrisome.

There are fancier Gaussian process toolsets. Chris Fonnesbeck mentions GPflow, autogp, PyMC3, and the scikit-learn implementation. Plus I notice skgmm is a fancified version of the scikit-learn one. George is another python GP regression that claims to handle big data at the cost of lots of C++. [GPStuff])https://github.com/gpstuff-dev/gpstuff) is the one for MATLAB/Octave that I have seen around the place. So… It’s easy enough to be bikeshedded is the message I’m getting here.

# Refs

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