Proposals which speed-up function-space MCMC

Inverse problems lend themselves naturally to a Bayesian formulation, in which the quantity of interest is a posterior distribution of state and/or parameters given some uncertain observations. For the common case in which the forward operator is smoothing, then the inverse problem is ill-posed. Well-posedness is imposed via regularisation in the form of a prior, which is often Gaussian. Under quite general conditions, it can be shown that the posterior is absolutely continuous with respect to the prior and it may be well-defined on function space in terms of its density with respect to the prior. In this case, by constructing a proposal for which the prior is invariant, one can define Metropolis-Hastings schemes for MCMC which are well-defined on function space, and hence do not degenerate as the dimension of the underlying quantity of interest increases to infinity, e.g. under mesh refinement when approximating PDE in finite dimensions. However, in practice, despite the attractive theoretical properties of the currently available schemes, they may still suffer from long correlation times, particularly if the data is very informative about some of the unknown parameters. In fact, in this case it may be the directions of the posterior which coincide with the (already known) prior which decorrelate the slowest. The information incorporated into the posterior through the data is often contained within some finite-dimensional subspace, in an appropriate basis, perhaps even one defined by eigenfunctions of the prior. We aim to exploit this fact and improve the mixing time of function-space MCMC by careful rescaling of the proposal. To this end, we introduce two new basic methods of increasing complexity, involving (i) characteristic function truncation of high frequencies and (ii) hessian information to interpolate between low and high frequencies

Deterministic Mean-field Ensemble Kalman Filtering

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $\kappa$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2\kappa$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory

Multilevel Particle Filters for L\'evy-driven stochastic differential equations

We develop algorithms for computing expectations of the laws of models associated to stochastic differential equations (SDEs) driven by pure L\'evy processes. We consider filtering such processes and well as pricing of path dependent options. We propose a multilevel particle filter (MLPF) to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions of the discretization of the underlying driving L\'evy proccess, our proposed method achieves optimal convergence rates. The cost to obtain MSE $O(\epsilon^2)$ scales like $O(\epsilon^{-2})$ for our method, as compared with the standard particle filter $O(\epsilon^{-3})$

Sparse online variational Bayesian regression

This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [65]. It also includes a family of priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for p unknown covariates the method can be implemented exactly online with a cost of $O(p^3)$ in computation and $O(p^2)$ in memory per iteration -- in other words, the cost per iteration is independent of n, and in principle infinite data can be considered. For large $p$ an approximation is able to achieve promising results for a cost of $O(p)$ per iteration, in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles $n = 65536$, $p = 131073$ on a laptop in less than 30 minutes, and $n = 10^5$, $p = 2.1 \times 10^6$ overnight

Determining White Noise Forcing From Eulerian Observations in the Navier Stokes Equation

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data. We complement this theoretical result with numerical simulation of the posterior distribution

Multilevel ensemble Kalman filtering for spatio-temporal processes

We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo (MLMC) sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficient hierarchical filtering method for spatio-temporal models. Under sufficient regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.Comment: Version 1: 39 pages, 4 figures.arXiv admin note: substantial text overlap with arXiv:1608.08558 . Version 2 (this version): 52 pages, 6 figures. Revision primarily of the introduction and the numerical examples sectio

A Bayesian analysis of classical shadows

The method of classical shadows heralds unprecedented opportunities for quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020)]. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations -- such as high-fidelity ground truth states -- which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions.Comment: 8 pages, 5 figure