This paper is concerned with sigma-point methods for filtering in nonlinear systems, where the process and measurement noise are heavy tailed and enter the system non-additively. The problem is approached within the framework of assumed density filtering and the necessary statistics are approximated using sigma-point methods developed for Student's t-distribution. This leads to UKF/CKF-type of fil

This letter presents the development of novel iterated filters and smoothers that only require specification of the conditional moments of the dynamic and measurement models. This leads to generalizations of the iterated extended Kalman filter, the iterated extended Kalman smoother, the iterated posterior linearization filter, and the iterated posterior linearization smoother. The connections to t

Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the data set. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless data set. Specifically, we consider the scenario where the

There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of ν times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducin

In this paper, we propose a new framework for solving state estimation problems with an additional sparsity-promoting $L-1$-regularizer term. We first formulate such problems as minimization of the sum of linear or nonlinear quadratic error terms and an extra regularizer, and then present novel algorithms which solve the linear and nonlinear cases. The methods are based on a combination of the ite

This paper proposes a novel algorithm for target tracking with direction-of-arrival measurements, modeled by von Mises-Fisher distributions. The algorithm makes use of the assumed density framework with Gaussian distributions, in which the posterior probability density of the target state is approximated by a Gaussian density. A key component of this algorithm is that the proposed Bayesian model o

In this letter, we analyze certain student's t-filters for linear Gaussian systems with misspecified noise covariances. It is shown that under appropriate conditions, the filter both estimates the state and re-scales the noise covariance matrices in a Kullback-Leibler optimal fashion. If the noise covariances are misscaled by a common scalar, then the re-scaling is asymptotically exact. We also co

This letter proposes a new algorithm for Gaussian process classification based on posterior linearization (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearization of the conditional mean of the labels and accounting for the linearization error. PL has some theoretical advantages over expectation propagation (EP): all calculated co

Owing to their millisecond-scale temporal resolution, magnetoencephalography (MEG) and electroencephalography (EEG) are well-suited tools to study dynamic functional connectivity between regions in the human brain. However, current techniques to estimate functional connectivity from MEG/EEG are based on a two-step approach; first, the MEG/EEG inverse problem is solved to estimate the source activi

This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on (1) taking the limit of statistical linear regression of the discretised process and (2) minimising an upper b

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss–Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyper-parameter estimation in Gauss–Markov regression, which tends to be considerably easier. The method’s relation

This article studies the maximum likelihood estimates of magnitude and scale parameters for a Gaussian process of Ornstein-Uhlenbeck type used to model a deterministic function that does not have to be a realisation of an Ornstein- Uhlenbeck process. Specifically, we derive explicit expressions for the limiting values of the maximum likelihood estimates as the number of observations increases. The

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, inste

In this letter, we consider Gaussian approximations of the optimal importance density in sequential importance sampling for nonlinear, non-Gaussian state-space models. The proposed method is based on generalized statistical linear regression and posterior linearization using conditional expectations. Simulation results show that the method outperforms the compared methods in terms of the effective

In this article, we address the problem of estimating the state and learning of the parameters in a linear dynamic system with generalized L 1 -regularization. Assuming a sparsity prior on the state, the joint state estimation and parameter learning problem is cast as an unconstrained optimization problem. However, when the dimensionality of state or parameters is large, memory requirements and co

In this paper, we develop a novel method for approximate continuous-discrete Bayesian filtering. The projection filtering framework is exploited to develop accurate approximations of posterior distributions within parametric classes of probability distributions. This is done by formulating an ordinary differential equation for the posterior distribution that has the prior as initial value and hits

In this paper, the connection between the Matérn kernel and scale mixtures of squared exponential kernels is explored. It is shown that the Matérn kernel can be approximated by a finite scale mixture of squared exponential kernels through a quadrature approximation which in turn allows for (i) state space approximations of the Matérn kernel for arbitrary smoothness parameters using established sta

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are reviewed, which agree with the classical results except that the formal adjoint of the generator is different in general. These re

Microelectromechanical-systems-based inertial sensors and magnetometers are low-cost, off-the-shelf sensors that are widely used in both consumer and industrial applications. However, these sensors suffer from biases and effects such as axis misalignment or scale errors, which require careful system design and periodic sensor calibration. In this paper, we propose a fast calibration method for joi

This paper is concerned with inferring the state of a Itô stochastic differential equation (SDE) from noisy discrete-time measurements. The problem is approached by considering basis function expansions of Brownian motion, that as a consequence give approximations to the underlying stochastic differential equation in terms of an ordinary differential equation with random coefficients. This allows