Densities separable in spherical coordinates have two advantages:$i$) the normalization constant is easy to compute, as the cumulative distribution can be decomposed into individual scalar integrals, and ii) an orthogonal inverse transform is directly available via a simple, scalar initial value problem and can be used to compute deterministic samples. We propagate uniform low-discrepancy sequences through that orthogonal inverse transform and obtain very homogeneous and even visually appealing deterministic samples. To demonstrate this technique, we exemplarily propose some spherical-coordinate-separable densities in$\mathbb{S}^{2}, \mathbb{R}^{2}$, and$\mathbb{R}^{3}$, including a non-isotropic modification of the von Mises-Fisher distribution. The proposed densities may be used, e.g., to represent uncertain radar measurements and for directional estimation. Furthermore, the framework presented herein allows quite simple design of various more densities tailored to a given scenario.

Markov chain Monte Carlo (MCMC) methods are a powerful but computationally expensive way of performing nonparametric Bayesian inference. MCMC proposals which utilise gradients, such as Hamiltonian Monte Carlo (HMC), can better explore the parameter space of interest if the additional hyperparameters are chosen well. The No-U-Turn Sampler (NUTS) is a variant of HMC which is extremely effective at selecting these hyper-parameters but is slow to run and is not suited to GPU architectures. An alternative to NUTS, Change in the Estimator of the Expected Square HMC (ChEES-HMC) was shown not only to run faster than NUTS on GPU but also sample from posteriors more efficiently. Sequential Monte Carlo (SMC) samplers are another sampling method which instead output weighted samples from the posterior. They are very amenable to parallelisation and therefore being run on GPUs while having additional flexibility in their choice of proposal over MCMC. We incorporate (ChEES-HMC) as a proposal into SMC samplers and demonstrate competitive but faster performance than NUTS on a number of tasks.

We propose a novel update step of a Gaussian mixture particle filter for nonlinear state estimation. The update procedure works as follows: First, unweighted samples are drawn in an optimal deterministic sense from a prior Gaussian mixture. These samples are then assigned weights from the likelihood function, and we compute higher-order moments from this samplebased posterior. These moment approximations converge with $L^{-1}$ instead of $L^{-1 / 2}$ as our samples are optimal deterministic. Finally, the continuous posterior approximation is determined as the Gaussian mixture that has minimal Fisher information under the constraint of having the aforementioned moments. To achieve this, we employ a closed-form solution of the Fisher information that involves Gaussian root mixture densities.

We present a new deterministic sampling method for arbitrary densities, unnormalized densities, and likelihoods. Our rejection-free and kernel-free method uses dense equal sphere packing of the volume under the density function (PoVuD). In order to obtain an ensemble that is better than independent random particles, we enforce some local homogeneity.

Nonlinear filtering is the most important aspect in state estimation with real-world systems. While the Kalman filter provides a simple though optimal estimate for linear systems, feasible filters for general systems are still subject of intensive research. The previously proposed Progressive Gaussian Filter PGF42 marked a new milestone, as it was able to efficiently compute an optimal Gaussian approximation of the posterior density in nonlinear systems [1]. However, for highly nonlinear systems where true posteriors are “banana-shaped” (e.g., cubic sensor problem) or multimodal (e.g., extended object tracking), even an optimal Gaussian approximation is an inadequate representation. Therefore, we generalize the established framework around the PGF42 from Gaussian to Gaussian mixture densities that are better able to approximate arbitrary density functions. Our filter simultaneously holds approximate Gaussian mixture and Dirac mixture representations of the same density, what we call coupled discrete and continuous densities (CoDiCo). For conversion between discrete and continuous representation, we employ deterministic sampling and the expectation-maximization (EM) algorithm, which we extend to deal with weighted particles.