forked from TH_General/Template_Summary
81 lines
5.2 KiB
TeX
81 lines
5.2 KiB
TeX
\chapter{Gaussian Identities}%
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\label{cha:Gaussian Identities}
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Im folgenden werden 3 Gaussian Identities genauer betrachtet
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\begin{enumerate}
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\item Marginal Gaussian Distribution: die Gaußsche Normalverteilung von einem Faktor
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\begin{equation} \label{eq:marginal_gaussian_distribution}
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p(\bm x) = \nomeq{gaussian_distribution}(\bm x|\bm \mu_x,\nomeq{covariance}_{\bm x})
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\end{equation}
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\item Conditional Gaussian Distribution: die Gaußsche Normalverteilung des zweiten Faktors in Abhängigkeit vom ersten Faktor
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($\bm F$: lineares Modell, dass $\bm x$ und $\bm y$ verknüpft)
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\begin{equation} \label{eq:conditional_gaussian_distribution}
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p(\bm y|\bm x) = \nomeq{gaussian_distribution}(\bm y|\bm F\bm x,\nomeq{covariance}_{\bm y})
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\end{equation}
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\item Joint Gaussian Distribution: Die Gaußsche Normalverteilung in Abhängigkeit von beiden Faktoren
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\begin{equation} \label{eq:joint_gaussian_distribution}
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p(\bm x,\bm y)
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= \nomeq{gaussian_distribution}\left( \begin{bmatrix}\bm x\\ \bm y \end{bmatrix} |
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\begin{bmatrix} \bm\mu_{\bm x} \\ \bm F\bm\mu_{\bm x} \end{bmatrix},
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\begin{bmatrix}
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\nomeq{covariance}_{\bm x} & \nomeq{covariance}_{\bm x}\bm F^T \\
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\bm F\nomeq{covariance}_{\bm x} & \nomeq{covariance}_{\bm y} + \bm F\nomeq{covariance}_{\bm x} \bm F^T
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\end{bmatrix} \right)
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\end{equation}
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\end{enumerate}
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Es gibt zwei Richtungen,
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in die umgewandelt werden kann:
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\begin{enumerate}
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\item Marginal und Conditional wird zu Joint:\\
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($\bm F$: lineares Modell, dass $\bm x$ und $\bm y$ verknüpft)
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\begin{equation} \label{eq:joint_gaussian_distribution_from_marginal_and_conditional}
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\nomeq{gaussian_distribution}(\bm x|\bm\mu_{\bm x},\nomeq{covariance}_{\bm x})\nomeq{gaussian_distribution}(\bm y|\bm F\bm x,\nomeq{covariance}_{\bm y})
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= \nomeq{gaussian_distribution}\left( \begin{bmatrix}\bm x\\ \bm y \end{bmatrix} |
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\begin{bmatrix} \bm\mu_{\bm x} \\ \bm F\bm\mu_{\bm x} \end{bmatrix},
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\begin{bmatrix}
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\nomeq{covariance}_{\bm x} & \nomeq{covariance}_{\bm x}\bm F^T \\
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\bm F\nomeq{covariance}_{\bm x} & \nomeq{covariance}_{\bm y} + \bm F\nomeq{covariance}_{\bm x} \bm F^T
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\end{bmatrix} \right)
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\end{equation}
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\item Joint zu Marginal und Conditional:\\
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($C$ ist wieder eine Matrix, die eine lineare Relation zwischen $x$ und $y$ herstellt)
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\begin{equation} \label{eq:marginal_and_conditional_gaussian_distribution_from_joint}
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\nomeq{gaussian_distribution}\left( \begin{bmatrix}\bm x\\ \bm y \end{bmatrix} |
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\begin{bmatrix} \bm\mu_{\bm x} \\ \bm\mu_{\bm y} \end{bmatrix},
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\begin{bmatrix} \nomeq{covariance}_{\bm x} & \bm C \\ \bm C^T & \nomeq{covariance}_{\bm y} \end{bmatrix} \right)
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= \nomeq{gaussian_distribution}(\bm x|\bm\mu_{\bm x},\nomeq{covariance}_{\bm x})
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\nomeq{gaussian_distribution}(\bm y| \bm\mu_{\bm y} + \bm C^T\nomeq{covariance}_{\bm x}^{-1}(\bm x - \bm\mu_{\bm x}),
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\nomeq{covariance}_{\bm y} - \bm C^T\nomeq{covariance}_{\bm x}^{-1}\bm C)
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\end{equation}
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\end{enumerate}
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\section{Gaussian Bayes Rules}%
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\label{sec:Gaussian Bayes Rules}
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Es gibt zwei bayesische Regeln für die Errechnung des Posteriors:\\
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(Herleitung: \cref{sec:Herleitung: Gaussian Bayes Rules})\\
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Gegeben: Marginal (\cref{eq:marginal_gaussian_distribution}) und Conditional (\cref{eq:conditional_gaussian_distribution})
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\begin{itemize}
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\item Gaussian Bayes Rule 1:
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\begin{itemize}
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\item Mean: \tabto{2.2cm}$\bm\mu_{\bm x|\bm y}
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= \bm\mu_{\bm x} + \nomeq{covariance}_{\bm x}\bm F^T(\sigma_{\bm y}^2\nomeq{identity_matrix} + \bm F\nomeq{covariance}_{\bm x}\bm F^T)^{-1}(\bm y - \bm F\bm\mu_{\bm x})$
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\item Covariance:\tabto{2.2cm} $\nomeq{covariance}_{\bm x|\bm y}
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= \nomeq{covariance}_{\bm x}-\nomeq{covariance}_{\bm x}\bm F^T(\sigma_{\bm y}^2\nomeq{identity_matrix}+\bm F\nomeq{covariance}_{\bm x}\bm F^T)^{-1}\bm F\nomeq{covariance}_{\bm x}$
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\end{itemize}
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\item Gaussian Bayes Rule 2:
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\begin{itemize}
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\item Mean: \tabto{2.2cm}$\bm\mu_{\bm x|\bm y}
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= \bm\mu_{\bm x} + (\sigma_{\bm y}^2\nomeq{covariance}_{\bm x}^{-1} + \bm F^T\bm F)^{-1} \bm F^T (\bm y - \bm F\bm\mu_{\bm x})$
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\item Covariance:\tabto{2.2cm} $\nomeq{covariance}_{\bm x|\bm y}
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= \sigma_{\bm y}^2(\sigma_{\bm y}^2\nomeq{covariance}_{\bm x}^{-1} + \bm F^T\bm F)^{-1}$
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\end{itemize}
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\end{itemize}
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\section{Gaussian Propagation}%
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\label{sec:Gaussian Propagation}
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Mit den Marginal und Conditional aus \cref{eq:marginal_gaussian_distribution} und \cref{eq:conditional_gaussian_distribution} ist es möglich den Conditional $p(\bm y)$ zu ermitteln:\\
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(Herleitung: \cref{sec:Herleitung: Gaussian Propagation})
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\begin{itemize}
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\item Mean: \tabto{2.2cm}$\bm\mu_{\bm y} = \bm F\bm\mu_{\bm x}$
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\item Covariance:\tabto{2.2cm} $\nomeq{covariance}_{\bm y} = \sigma_{\bm y}^2\nomeq{identity_matrix} + \bm F\nomeq{covariance}_{\bm x}\bm F^T$
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\end{itemize}
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