%appendix
\appendix
\chapter{Anhang}
\label{appendix}

\section{Herleitung: Gradient for Logistic Regression}%
\label{sec:Herleitung: Gradient for Logistic Regression}
\includegraphics[page=64,width=\textwidth]{Vorlesungen/02_LinearClassification.pdf}

\section{Herleitung: Multiclass Classification: Data log-likelihood}%
\label{sec:Herleitung: Multiclass Classification: Data log-likelihood}
\includegraphics[page=68,width=\textwidth]{Vorlesungen/02_LinearClassification.pdf}

\section{Herleitung: CART: Classification Tree}%
\label{sec:Herleitung: CART: Classification Tree}
\includegraphics[page=32,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=33,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=34,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=35,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=36,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}

\section{Herleitung: CART: Regression Tree}%
\label{sec:Herleitung: CART: Regression Tree}
\includegraphics[page=24,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=25,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=26,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=27,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=28,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=29,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=30,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}
\includegraphics[page=31,width=\textwidth]{Vorlesungen/04_TreesAndForests.pdf}

\section{Herleitung: Soft Max-Margin: Hinge Loss}%
\label{sec:Herleitung: Soft Max-Margin: Hinge Loss}
\includegraphics[page=21,width=\textwidth]{Vorlesungen/06_SVMs.pdf}

\section{Anwendungsbeispiele: \glstopshortpl{SVM}}%
\label{sec:Anwendungsbeispiele: SVMs}
\includegraphics[page=34,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=35,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=36,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=37,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=38,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=39,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=40,width=\textwidth]{Vorlesungen/06_SVMs.pdf}

\section{Herleitung: SVMs with Kernels}%
\label{sec:Herleitung: SVMs with Kernels}
\includegraphics[page=52,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=53,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=54,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=55,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=56,width=\textwidth]{Vorlesungen/06_SVMs.pdf}

\section{Beispiele: SVM: Model Selection}%
\label{sec:Beispiele: SVM: Model Selection}
\includegraphics[page=57,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=58,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=59,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=60,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=62,width=\textwidth]{Vorlesungen/06_SVMs.pdf}
\includegraphics[page=63,width=\textwidth]{Vorlesungen/06_SVMs.pdf}

\section{Anwendungsbeispiel: Bayesian Learning: Regression}%
\label{sec:Anwendungsbeispiel: Bayesian Learning: Regression}
\includegraphics[page=18,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=19,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=20,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Beweis: Gaussian Processes ist eine kernelized Bayesian Linear Regression}%
\label{sec:Beweis: Gaussian Processes ist eine kernelized Bayesian Linear Regression}
\includegraphics[page=41,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=42,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=43,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=44,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=45,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Herleitung: Gaussian Processes: Posterior}%
\label{sec:Herleitung: Gaussian Processes: Posterior}
\includegraphics[page=38,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Herleitung: Gaussian Processes: \nomsym{mean} und \nomsym{variance}}%
\label{sec:Herleitung: Gaussian Processes: mean and variance}
\includegraphics[page=39,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Beispiel: Neural Network: XOR}%
\label{sec:Beispiel: Neural Network: XOR}
\includegraphics[page=25,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}
\includegraphics[page=26,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}

\section{Beispiel: Neural Networks: Feature Learning}%
\label{sec:Beispiel: Neural Networks: Feature Learning}
\includegraphics[page=35,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}

\section{Herleitung: Backpropagation in Matrix-Form}%
\label{sec:Herleitung: Backpropagation in Matrix-Form}
\includegraphics[page=52,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}
\includegraphics[page=53,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}

\section{Zusätzliche Informationen: Second Order Optimization}%
\label{sec:Zusaetzliche Informationen: Second Order Optimization}
\includegraphics[page=74,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}
\includegraphics[page=75,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}
\includegraphics[page=76,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}

\section{Zusätzliche Informationen: MNIST Datensatz}%
\label{sec:Zusaetzliche Informationen: MNIST Datensatz}
\includegraphics[page=82,width=\textwidth]{Vorlesungen/08_NeuralNets.pdf}

\section{Anwendungsbeispiele für CNNs}%
\label{sec:Anwendungsbeispiele fuer CNNs}
\includegraphics[page=3,width=\textwidth]{Vorlesungen/09_CNNs+RNNs.pdf}
\includegraphics[page=4,width=\textwidth]{Vorlesungen/09_CNNs+RNNs.pdf}

\section{Beispiel: Convolutional Layer: Stride and Padding}%
\label{sec:Beispiel: Convolutional Layer: Stride and Padding}
\includegraphics[page=14,width=\textwidth]{Vorlesungen/09_CNNs+RNNs.pdf}
\includegraphics[page=15,width=\textwidth]{Vorlesungen/09_CNNs+RNNs.pdf}

\section{Herleitung: Dimensionality Reduction: Minimizing the Error}%
\label{sec:Herleitung: Dimensionality Reduction: Minimizing the Error}
\includegraphics[page=16,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}

\section{Herleitung: PCA: Maximierungsproblem in Matrix-Schreibweise}%
\label{sec:Herleitung: PCA: Maximierungsproblem in Matrix-Schreibweise}
\includegraphics[page=19,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}

\section{Anwendungsbeispiele: PCA}%
\label{sec:Anwendungsbeispiele: PCA}
\includegraphics[page=27,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=28,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=29,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=30,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=31,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=32,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}
\includegraphics[page=33,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}

\section{Beweis: K-Means Konvergenz}%
\label{sec:Beweis: K-Means Konvergenz}
\includegraphics[page=49,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}

\section{Formale Definition: Histrograms}%
\label{sec:Formale Definition: Histrograms}
\includegraphics[page=64,width=\textwidth]{Vorlesungen/10_DimensionalityReductionClustering.pdf}

\section{Herleitung: Differenzierung des \glstopshortpl{GMM}}%
\label{sec:Herleitung: Differenzierung des GMMs}
\includegraphics[page=7,width=\textwidth]{Vorlesungen/11 - ExpectationMaximization.pdf}

\section{Herleitung: \glstopshort{EM}-Decomposition}%
\label{sec:Herleitung: EM-Decomposition}
\includegraphics[page=24,width=\textwidth]{Vorlesungen/11 - ExpectationMaximization.pdf}

\section{Herleitung: EM for GMMs: Maximization"~Step}%
\label{sec:Herleitung: EM for GMMs: Maximization-Step}
\includegraphics[page=16,width=\textwidth]{Vorlesungen/11 - ExpectationMaximization.pdf}

\section{Herleitung: EM for Dimensionality Reduction: Maximization"~Step}%
\label{sec:Herleitung: EM for Dimensionality Reduction: Maximization-Step}
\includegraphics[page=38,width=\textwidth]{Vorlesungen/11 - ExpectationMaximization.pdf}

\section{Herleitung: EM for Dimensionality Reduction: Maximization"~Step: Monte-Carlo Esitmation}%
\label{sec:Herleitung: EM for Dimensionality Reduction: Maximization-Step: Monte-Carlo Esitmation}
\includegraphics[page=40,width=\textwidth]{Vorlesungen/11 - ExpectationMaximization.pdf}

\section{Herleitung: Variational Bayes: Maximierung des Marginal Log"~Likelihood}%
\label{sec:Herleitung: Variational Bayes: Maximierung des Marginal Log-Likelihood}
\includegraphics[page=13,width=\textwidth]{Vorlesungen/12 - VaraitionalAutoEncoders.pdf}
\includegraphics[page=14,width=\textwidth]{Vorlesungen/12 - VaraitionalAutoEncoders.pdf}

\section{Reparameterization Trick}%
\label{sec:Reparameterization Trick}
\includegraphics[page=19,width=\textwidth]{Vorlesungen/12 - VaraitionalAutoEncoders.pdf}
\includegraphics[page=20,width=\textwidth]{Vorlesungen/12 - VaraitionalAutoEncoders.pdf}

\section{Zusätzliche Informationen: Optimization over the variational distribution}%
\label{sec:Zusaetzliche Informationen: Optimization over the variational distribution}
\includegraphics[page=21,width=\textwidth]{Vorlesungen/12 - VaraitionalAutoEncoders.pdf}

\section{Zusätzliche Informationen: MLE: conditional log-likelihood}%
\label{sec:Zusaetzliche Informationen: MLE: conditional log-likelihood}
\includegraphics[page=21,width=\textwidth]{Vorlesungen/02_LinearClassification.pdf}
\includegraphics[page=22,width=\textwidth]{Vorlesungen/02_LinearClassification.pdf}

\section{Beweis für die positive Definitheit des Gaussian Kernels}%
\label{sec:Beweis fuer die positive Definitheit des Gaussian Kernels}
\includegraphics[page=14,width=\textwidth]{Vorlesungen/05_KernelMethods.pdf}
\includegraphics[page=15,width=\textwidth]{Vorlesungen/05_KernelMethods.pdf}

\section{Beispiele für die Optimierung von Hyper-Parametern eines Gaussian Kernels}%
\label{sec:Beispiele fuer die Optimierung von Hyper-Parametern eines Gaussian Kernels}
\includegraphics[page=53,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=54,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=55,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=56,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=57,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Herleitung: Gaussian Bayes Rules}%
\label{sec:Herleitung: Gaussian Bayes Rules}
\includegraphics[page=26,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}
\includegraphics[page=27,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}

\section{Herleitung: Gaussian Propagation}%
\label{sec:Herleitung: Gaussian Propagation}
\includegraphics[page=29,width=\textwidth]{Vorlesungen/07_BayesianLearning.pdf}


