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By restricting to Gaussian distributions, the optimal Bayesian filtering problem can be transformed into an algebraically simple form, which allows for computationally efficient algorithms. Three problem settings are discussed in this thesis: (1) filtering with Gaussians only, (2) Gaussian mixture filtering for strong nonlinearities, (3) Gaussian process filtering for purely data-driven scenarios. For each setting, efficient algorithms are derived and applied to real-world problems. Background & Summary Introduction Nonlinear Bayesian Filtering Dynamic Models and Measurement Models Recursive Filtering Closed-form Calculation Approximate Filtering: State of the Art Research Topics Main Contributions Gaussian Filtering Gaussian Mixture Filtering Gaussian Process Filtering Thesis Outline Gaussian Filtering The Gaussian Distribution Importance of the Gaussian Dirac Delta Distribution The Exponential Family Exact Gaussian Filtering and Approximations General Formulation Linear Filtering Linearized and Extended Kalman Filter Statistical Linearization Linear Regression Kalman Filters Gaussian Smoothing General Formulation Linear Case Nonlinear Case Rao-Blackwellization Contributions Combining Rao-Blackwellization with Observed-Unobserved Decomposition Semi-Analytical Filtering Chebyshev Polynomial Kalman Filtering Efficient Moment Propagation for Polynomials Homotopic Moment Matching for Polynomial Measurement Models Summary Gaussian Mixture Filtering Gaussian Mixtures Nonlinear Filtering Individual Approximation Generic Gaussian Mixture Filter Component Adaptation Weight Optimization Reduction Refinement Contributions Semi-Analytic Gaussian Mixture Filter Adaptive Gaussian Mixture Filter Curvature-based Gaussian Mixture Reduction Summary Gaussian Process Filtering Gaussian Processes Covariance Functions Examples Hyperparameter Learning Large Data Sets Active Set Approaches Local Approaches Algebraic Tricks Open Issues Nonlinear Filtering Contributions Gaussian Process Filtering Gaussian Process Smoothing Recursive Gaussian Process Regression On-line Hyperparameter Learning Summary Applications Range-based Localization Position Estimation Position and Orientation Estimation Gas Dispersion Source Estimation Atmospheric Dispersion Models Parameter Estimation Active Object Recognition Object Classification Learning Estimation Planning Summary Concluding Remarks Conclusions Future Work Particle Filtering Perfect Monte Carlo Sampling Importance Sampling Sequential Importance Sampling Choice of Importance Function Resampling Performance Measures Root Mean Square Error Mean Absolute Error Normalized Estimation Error Square Negative Log-Likelihood Quadratic Programming Bibliography Publications Gaussian Filtering using State Decomposition Methods Semi-Analytic Gaussian Assumed Density Filter Chebyshev Polynomial Kalman Filter Gaussian Filtering for Polynomial Systems (Semi-)Analytic Gaussian Mixture Filter Adaptive Gaussian Mixture Filter Superficial Gaussian Mixture Reduction Analytic Moment-based Gaussian Process Filtering Robust Filtering and Smoothing with Gaussian Processes Recursive Gaussian Process Regression Recursive Gaussian Process: On-line Regression and Learning Optimal Stochastic Linearization for Range-Based Localization Semi-Analytic Stochastic Linearization for Pose Tracking On-line Dispersion Source Estimation Bayesian Active Object Recognition