Gaussian Process

[[concept]]

[!themes] Topics

Evaluation Error: SyntaxError: Unexpected token '>'

at DataviewInlineApi.eval (plugin:dataview:19027:21)

at evalInContext (plugin:dataview:19028:7)

at asyncEvalInContext (plugin:dataview:19038:32)

at DataviewJSRenderer.render (plugin:dataview:19064:19)

at DataviewJSRenderer.onload (plugin:dataview:18606:14)

at DataviewJSRenderer.load (app://obsidian.md/app.js:1:1182416)

at DataviewApi.executeJs (plugin:dataview:19607:18)

at DataviewCompiler.eval (plugin:digitalgarden:10763:23)

at Generator.next (<anonymous>)

at eval (plugin:digitalgarden:90:61)

Definition

Bayesian Statistics

Gaussian Process

A Gaussian Process is a probability distribution over all $y (x) : X \to R$ such that $y (x)$ evaluated at any of $N$ points $x_{1}, x_{2}, \dots, x_{N}$ has a multivariate normal distribution.

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

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

Let $y_{1}, \dots, y_{n} \in R^{m}$ and $G \sim N (0, 1)^{\otimes d \times m}$ . Then
$Law ([\begin{matrix} G y_{1} \\ ⋮ \\ G y_{n} \end{matrix}]) = N (0, [\begin{matrix} | | y_{1} | |^{2} I_{d} & ⟨ y_{1}, y_{2} ⟩ I_{d} & \dots & ⟨ y_{1}, y_{n} ⟩ I_{d} \\ ⟨ y_{2}, y_{1} ⟩ I_{d} & | | y_{2} | |^{2} I_{d} & \dots & ⟨ y_{2}, y_{n} ⟩ I_{d} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ⟨ y_{n}, y_{1} ⟩ I_{d} & ⟨ y_{n}, y_{2} ⟩ I_{d} & \dots & | | y_{n} | |^{2} I_{d} \end{matrix}])$
Or, if we define $Y$ such that the $i$ th column is $y_{i}$ , then the the covariance matrix is given by

$\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 $G y \in R^{d}$ to each fixed vector $y \in R^{m}$ .

References

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}

Gaussian Process

Definition

Bayesian Statistics

Random Matrix Theory

References

See Also

Mentions