The expressive power of Bayesian kernel-based methods has led them to become an important tool across many different facets of artificial intelligence, and useful to a plethora of modern application domains, providing both power and interpretability via uncertainty analysis. This article introduces and discusses two methods which straddle the areas of probabilistic Bayesian schemes and kernel methods for regression: Gaussian Processes and Relevance Vector Machines. Our focus is on developing a common framework with which to view these methods, via intermediate methods a probabilistic version of the well-known kernel ridge regression, and drawing connections among them, via dual formulations, and discussion of their application in the context of major tasks: regression, smoothing, interpolation, and filtering. Overall, we provide understanding of the mathematical concepts behind these models, and we summarize and discuss in depth different interpretations and highlight the relationship to other methods, such as linear kernel smoothers, Kalman filtering and Fourier approximations. Throughout, we provide numerous figures to promote understanding, and we make numerous recommendations to practitioners. Benefits and drawbacks of the different techniques are highlighted. To our knowledge, this is the most in-depth study of its kind to date focused on these two methods, and will be relevant to theoretical understanding and practitioners throughout the domains of data-science, signal processing, machine learning, and artificial intelligence in general.
KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics
ETH Zürich（苏黎世联邦理工学院）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Eindhoven University of Technology（埃因霍温理工大学）；Phillips Research（飞利浦研究院）；Weizmann Institute of Science（魏茨曼科学研究所）
Outlier-robust Kalman Filtering through Generalised Bayes
School of Mathematical Sciences, Queen Mary University（伦敦玛丽大学数学科学学院）；Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学牛津-曼量化金融研究所）；Department of Statistical Science, University College London（伦敦大学学院统计科学系）；Department of Mathematics and Statistics, Memorial University of Newfoundland（纽芬兰纪念大学数学与统计系）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Institute for Cognitive Science, University of Colorado Boulder（科罗拉多大学博尔德分校认知科学研究所）；Google DeepMind（谷歌DeepMind）
ETH Zürich（苏黎世联邦理工学院）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Eindhoven University of Technology（埃因霍温理工大学）；Phillips Research（飞利浦研究院）；Weizmann Institute of Science（魏茨曼科学研究学院）
Uncertainty in Data-Driven Kalman Filtering for Partially Known State-Space Models
University of Haifa（海法大学）；ETH Zürich（苏黎世联邦理工学院）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Eindhoven University of Technology（埃因霍温理工大学）；Phillips Research（飞利浦研究院）；Weizmann Institute of Science（魏茨曼科学研究所）
Differentiable Adaptive Kalman Filtering via Optimal Transport
Shanghai Research Institute for Intelligent Autonomous Systems（上海智能自主系统研究所）；Tongji University（同济大学）；School of Computer Science and Technology（计算机科学与技术学院）；Shanghai Jiaotong University（上海交通大学）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究所）；East China Normal University（华东师范大学）
