2006 AdvancesinGaussianProcesses

(Rasmussen, 2006) ⇒ Carl Edward Rasmussen. (2006). “Advances in Gaussian Processes.” Tutorial at Advances in Neural Information Processing Systems, 19 (NIPS 2016).

Subject Headings: Gaussian Process Model, Gaussian Process Regression.

Notes

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Quotes

The Gaussian Distribution

[math]\displaystyle{ \mathcal{p}(\mathbf{x}|µ, \Sigma) = \mathcal{N}(µ,\Sigma) = (2\pi)^{-D/2}|\Sigma|^{-1/2} \exp\bigl(-\frac{1}{2}(x - µ))^T \Sigma^{-1}(x - µ)\bigr) }[/math]

where [math]\displaystyle{ µ }[/math] is the mean vector and [math]\displaystyle{ \Sigma }[/math] the covariance matrix.

Both the conditionals and the marginals of a joint Gaussian are again Gaussian.

What is a Gaussian Process?

A Gaussian process is a generalization of a multivariate Gaussian distribution to infinitely many variables.

Informally: infinitely long vector [math]\displaystyle{ \simeq }[/math] function

Definition: a Gaussian process is a collection of random variables, any finite number of which have (consistent) Gaussian distributions. �

A Gaussian distribution is fully specified by a mean vector, µ, and covariance matrix [math]\displaystyle{ \Sigma }[/math]:

[math]\displaystyle{ \mathbf{f} = (f_1,...,f_n)^T \sim \mathcal{N}(µ, \Sigma), \text{indexes} i = 1,...,n }[/math]

A Gaussian process is fully specified by a mean function m(x) and covariance function k(x,x'):

[math]\displaystyle{ f(x) \sim \mathcal{GP}(m(x), k(x,x')), \text{indexes} x }[/math]

…

References

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	Author	volume	Date Value	title	type	journal	titleUrl	doi	note	year
2006 AdvancesinGaussianProcesses	Carl Edward Rasmussen			Advances in Gaussian Processes						2006