Gaussian Process

[[concept]]

Topics

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Definition

Bayesian Statistics

Gaussian Process

A Gaussian Process is a probability distribution over all $y(x):X\to \mathbb{R}$ such that $y(x)$ evaluated at any of $N$ points $x_1,x_2,\dots ,x_N$ has a multivariate normal distribution.

If $x\in {\mathbb{R}}^2$, then the gaussian process is known as a Gaussian Random Field

Random Matrix Theory

The joint of gaussian random transform of finite vectors is the characterization of a Gaussian random field, which is a term for a high(er)-dimensional Gaussian Process.

Characterization

Proposition $y_1,\dots ,y_n\in {\mathbb{R}}^m$ and $G\sim \mathcal{N}(0,1{)}^{\otimes d\times m}$ . Then \text{Law}\left(\begin{bmatrix} Gy_{1} \\ \vdots \\ Gy_{n} \end{bmatrix}\right) = {\cal N}\left(0,\, \begin{bmatrix} \lvert \lvert y_{1} \rvert \rvert ^2 I_{d} & \langle y_{1},y_{2} \rangle I_{d} & \dots & \langle y_{1},y_{n} \rangle I_{d} \\ \langle y_{2} , y_{1} \rangle I_{d} & \lvert \lvert y_{2} \rvert \rvert ^2 I_{d} & \dots & \langle y_{2}, y_{n} \rangle I_{d} \\ \vdots & \vdots & \ddots & \vdots \\ \langle y_{n}, y_{1} \rangle I_{d} & \langle y_{n} , y_{2} \rangle I_{d} & \dots & \lvert \lvert y_{n} \rvert \rvert ^2 I_{d} \end{bmatrix}\right)$$ Or, if we define Y$suchthatthe$i$thcolumnis$y_{i}$, then the the covariance matrix is given by

Let

$\text{Cov}\left(\begin{bmatrix} Gy_{1} \ \vdots \ Gy_{n} \end{bmatrix}\right) = Y^{\intercal}Y \otimes I_{d} $$

In particular, this characterization associates the random vector $Gy\in {\mathbb{R}}^d$ to each fixed vector $y\in {\mathbb{R}}^m$.

References

References

See Also

Mentions

Mentions

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