Gaussian process: a standalone construct

Norbert Wiener's 1923 paper Differential-space constructed the first rigorous example of what is now called a Gaussian process, the random continuous path known as Brownian motion, whose values at any collection of times follow a joint Gaussian, or normal, distribution. The same mathematical object was rediscovered for prediction in geostatistics by Georges Matheron in 1963, whose method of kriging estimates an unknown value at a location from nearby measurements together with a measure of how uncertain that estimate is. Carl Rasmussen and Christopher Williams gave the modern cross-disciplinary treatment in their 2006 book Gaussian Processes for Machine Learning, defining a Gaussian process as a distribution over functions in which any finite set of function values is jointly Gaussian, fully specified by a mean function and a covariance (kernel) function. What makes the construct a landmark is that it turns curve fitting into probabilistic inference: rather than returning a single fitted function it returns a whole probability distribution over functions, so every prediction carries a calibrated estimate of its own uncertainty, and the choice of covariance function encodes assumptions such as smoothness or periodicity.[1][2][3]

Unit

A stochastic process, meaning a collection of random variables indexed by inputs, whose every finite subset has a joint Gaussian distribution

Recovery means

Not applicable

Measured by

Specified by a chosen mean function and covariance (kernel) function, then fitted to data by conditioning on the observed points, with the kernel's parameters typically set by maximising the marginal likelihood