Bayesian state estimation in dynamical systems is one of the key challenges in signal processing. The key idea is to obtain a (posterior) probability distribution over an unobserved (state) variable, based on noisy measurements. The state typically evolves according to a Markov chain. Except for the linear case (with Gaussian noise), no closed-form solutions to the state estimation problem are known. In this project, we focus on Bayesian state estimation in various nonlinear systems.
A General Perspective on Gaussian Filtering and Smoothing
We present a general probabilistic perspective on Gaussian filtering and smoothing. This allows us to show that common approaches to Gaussian filtering/smoothing can be distinguished solely by their methods of computing/approximating the means and covariances of joint probabilities. This implies that novel filters and smoothers can be derived straightforwardly by providing methods for computing these moments. Based on this insight, we derive the cubature Kalman smoother and propose a novel robust filtering and smoothing algorithm based on Gibbs sampling.
Filtering and Smoothing with Gaussian Processes
We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non-parametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning, robotics, and control for representing unknown system functions by posterior probability distributions. This modern way of “system identification” is more robust than finding point estimates of a parametric function representation. In this article, we present a principled algorithm for robust analytic smoothing in GP dynamic systems, which are increasingly used in robotics and control. Our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state-of-the-art Gaussian filters and smoothers can fail.
Message Passing in Nonlinear Dynamical Systems
Rich and complex time-series data, such as those generated from engineering systems, financial markets, videos, or neural recordings are now a common feature of modern data analysis. Explaining the phenomena underlying these diverse data sets requires flexible and accurate models. In this paper, we promote Gaussian process dynamical systems as a rich model class that is appropriate for such an analysis. We present a new approximate message-passing algorithm for Bayesian state estimation and inference in Gaussian process dynamical systems, a non- parametric probabilistic generalization of commonly used state-space models. We derive our message-passing algorithm using Expectation Propagation and provide a unifying perspective on message passing in general state-space models. We show that existing Gaussian filters and smoothers appear as special cases within our inference framework, and that these existing approaches can be improved upon using iterated message passing. Using both synthetic and real-world data, we demonstrate that iterated message passing can improve inference in a wide range of tasks in Bayesian state estimation, thus leading to improved predictions and more effective decision making.
Multi-Modal Filtering and Smoothing
In this project, we are interested in Bayesian state estimation with multi-modal density representations. Standard filters and smoothers (EKF, UKF etc.) usually approximate densities by Gaussians. Sometimes, these unimodal representations are insufficient to capture important properties of the underlying distribution, such as bifurcations. We propose a filtering algorithm that uses Gaussian mixture representations of densities to address this issue. Our method allows for analytic computation of the parameters of the mixture components and does not suffer from the common problem of under-estimating predictive uncertainties, which happens frequently in the EKF and the UKF.
Shakir Mohamed, University of British Columbia
Henrik Ohlsson, University of Linkoping
Ryan Turner, University of Cambridge
Marco Huber, Karlsruhe Institute of Technology
Sanket Kamthe, TU Darmstadt
Uwe D. Hanebeck, Karlsruhe Institute of Technology
Carl Edward Rasmussen, University of Cambridge