Robust Gaussian Filtering using a Pseudo Measurement

Many sensors, such as range, sonar, radar, GPS and visual devices, produce measurements which are contaminated by outliers. This problem can be addressed by using fat-tailed sensor models, which account for the possibility of outliers. Unfortunately, all estimation algorithms belonging to the family of Gaussian filters (such as the widely-used extended Kalman filter and unscented Kalman filter) are inherently incompatible with such fat-tailed sensor models. The contribution of this paper is to show that any Gaussian filter can be made compatible with fat-tailed sensor models by applying one simple change: Instead of filtering with the physical measurement, we propose to filter with a pseudo measurement obtained by applying a feature function to the physical measurement. We derive such a feature function which is optimal under some conditions. Simulation results show that the proposed method can effectively handle measurement outliers and allows for robust filtering in both linear and nonlinear systems

Robust Inference for State-Space Models with Skewed Measurement Noise

Filtering and smoothing algorithms for linear discrete-time state-space models with skewed and heavy-tailed measurement noise are presented. The algorithms use a variational Bayes approximation of the posterior distribution of models that have normal prior and skew-t-distributed measurement noise. The proposed filter and smoother are compared with conventional low-complexity alternatives in a simulated pseudorange positioning scenario. In the simulations the proposed methods achieve better accuracy than the alternative methods, the computational complexity of the filter being roughly 5 to 10 times that of the Kalman filter.Comment: 5 pages, 7 figures. Accepted for publication in IEEE Signal Processing Letter

A variational Bayesian method for inverse problems with impulsive noise

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy

t-Exponential Memory Networks for Question-Answering Machines

Recent advances in deep learning have brought to the fore models that can make multiple computational steps in the service of completing a task; these are capable of describ- ing long-term dependencies in sequential data. Novel recurrent attention models over possibly large external memory modules constitute the core mechanisms that enable these capabilities. Our work addresses learning subtler and more complex underlying temporal dynamics in language modeling tasks that deal with sparse sequential data. To this end, we improve upon these recent advances, by adopting concepts from the field of Bayesian statistics, namely variational inference. Our proposed approach consists in treating the network parameters as latent variables with a prior distribution imposed over them. Our statistical assumptions go beyond the standard practice of postulating Gaussian priors. Indeed, to allow for handling outliers, which are prevalent in long observed sequences of multivariate data, multivariate t-exponential distributions are imposed. On this basis, we proceed to infer corresponding posteriors; these can be used for inference and prediction at test time, in a way that accounts for the uncertainty in the available sparse training data. Specifically, to allow for our approach to best exploit the merits of the t-exponential family, our method considers a new t-divergence measure, which generalizes the concept of the Kullback-Leibler divergence. We perform an extensive experimental evaluation of our approach, using challenging language modeling benchmarks, and illustrate its superiority over existing state-of-the-art techniques

The Extended Parameter Filter

The parameters of temporal models, such as dynamic Bayesian networks, may be modelled in a Bayesian context as static or atemporal variables that influence transition probabilities at every time step. Particle filters fail for models that include such variables, while methods that use Gibbs sampling of parameter variables may incur a per-sample cost that grows linearly with the length of the observation sequence. Storvik devised a method for incremental computation of exact sufficient statistics that, for some cases, reduces the per-sample cost to a constant. In this paper, we demonstrate a connection between Storvik's filter and a Kalman filter in parameter space and establish more general conditions under which Storvik's filter works. Drawing on an analogy to the extended Kalman filter, we develop and analyze, both theoretically and experimentally, a Taylor approximation to the parameter posterior that allows Storvik's method to be applied to a broader class of models. Our experiments on both synthetic examples and real applications show improvement over existing methods

Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models

We propose a new class of filtering and smoothing methods for inference in high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including on-line and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods

Chance, long tails, and inference: a non-Gaussian, Bayesian theory of vocal learning in songbirds

Traditional theories of sensorimotor learning posit that animals use sensory error signals to find the optimal motor command in the face of Gaussian sensory and motor noise. However, most such theories cannot explain common behavioral observations, for example that smaller sensory errors are more readily corrected than larger errors and that large abrupt (but not gradually introduced) errors lead to weak learning. Here we propose a new theory of sensorimotor learning that explains these observations. The theory posits that the animal learns an entire probability distribution of motor commands rather than trying to arrive at a single optimal command, and that learning arises via Bayesian inference when new sensory information becomes available. We test this theory using data from a songbird, the Bengalese finch, that is adapting the pitch (fundamental frequency) of its song following perturbations of auditory feedback using miniature headphones. We observe the distribution of the sung pitches to have long, non-Gaussian tails, which, within our theory, explains the observed dynamics of learning. Further, the theory makes surprising predictions about the dynamics of the shape of the pitch distribution, which we confirm experimentally