Smoothing and filtering with a class of outer measures

Filtering and smoothing with a generalised representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalised Bayesian-like update equations. This leads to a system of generalised smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found in the classical Bayesian filtering framework. It is additionally shown that the Kalman filter recursion can be recovered from weaker assumptions on the available information on the corresponding hidden Markov model

An Introduction to Wishart Matrix Moments

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities

Controlled Sequential Monte Carlo

Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques for approximating high-dimensional probability distributions and their normalizing constants. These methods have found numerous applications in statistics and related fields; e.g. for inference in non-linear non-Gaussian state space models, and in complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which crucially impact their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are determined by approximating the solution to an associated optimal control problem using an iterative scheme. This method builds upon a number of existing algorithms in econometrics, physics, and statistics for inference in state space models, and generalizes these methods so as to accommodate complex static models. We provide a theoretical analysis concerning the fluctuation and stability of this methodology that also provides insight into the properties of related algorithms. We demonstrate significant gains over state-of-the-art methods at a fixed computational complexity on a variety of applications

A Multilevel Approach for Stochastic Nonlinear Optimal Control

We consider a class of finite time horizon nonlinear stochastic optimal control problem, where the control acts additively on the dynamics and the control cost is quadratic. This framework is flexible and has found applications in many domains. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging Monte Carlo task. The focus of this article is to propose a new Monte Carlo approach that significantly improves upon existing methodology. Our proposed methodology first tackles the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state space model associated with the control problem, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce computational cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with $\mathcal{O}(\epsilon^2)$ mean squared error with a computational cost of $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In contrast, a computational cost of $\mathcal{O}(\epsilon^{-3})$ is required for existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples, which validate our theory

On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering. We provide intuition for how these results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence

Variational Bayesian Model Selection for Mixture Distributions

Mixture models, in which a probability distribu-tion is represented as a linear superposition of component distributions, are widely used in sta-tistical modeling and pattern recognition. One of the key tasks in the application of mixture models is the determination of a suitable number of components. Conventional approaches based on cross-validation are computationally expen-sive, are wasteful of data, and give noisy esti-mates for the optimal number of components. A fully Bayesian treatment, based on Markov chain Monte Carlo methods for instance, will re-turn a posterior distribution over the number of components. However, in practical applications it is generally convenient, or even computation-ally essential, to select a single, most appropri-ate model. Recently it has been shown, in the context of linear latent variable models, that the use of hierarchical priors governed by continuous hyperparameters whose values are set by type-II maximum likelihood, can be used to optimize model complexity. In this paper we extend this framework to mixture distributions by consider-ing the classical task of density estimation us-ing mixtures of Gaussians. We show that, by setting the mixing coefficients to maximize the marginal log-likelihood, unwanted components can be suppressed, and the appropriate number of components for the mixture can be determined in a single training run without recourse to cross-validation. Our approach uses a variational treat-ment based on a factorized approximation to the posterior distribution.

Multi-time delay, multi-point Linear Stochastic Estimation of a cavity shear layer velocity from wall-pressure measurements

Multi-time-delay Linear Stochastic Estimation (MTD-LSE) technique is thoroughly described, focusing on its fundamental properties and potentialities. In the multi-time-delay ap- proach, the estimate of the temporal evolution of the velocity at a given location in the flow field is obtained from multiple past samples of the unconditional sources. The technique is applied to estimate the velocity in a cavity shear layer flow, based on wall-pressure measurements from multiple sensor