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High-dimensional regression in practice: an empirical study of finite-sample prediction, variable selection and ranking

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specif...

Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, si...

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for ...

The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as \(n^{-1/2}\). However, the re-weighting of random points, as exemplified in the Bayesian Monte Carlo method, can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor...

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g. Gaussian and Hardy kernels...

This article attempts to place the emergence of probabilistic numerics as a mathematical–statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul\('\)din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysi...

When models are defined implicitly by systems of differential equations with no closed-form solution, small local errors in finite-dimensional solution approximations can propagate into deviations from the true underlying model trajectory. Some recent perspectives in quantifying this uncertainty are based on Bayesian probability modeling: a prior is defined over the unknown solution and updated by conditioning on interrogations of the forward model. Improvement in accuracy via grid refinement mu...

Donoho and Johnstone (Ann Stat 26(3):879–921, 1998) proposed a method from reconstructing an unknown smooth function u from noisy data \(u+\zeta \) by translating the empirical wavelet coefficients of \(u+\zeta \) towards zero. We consider the situation where the prior information on the unknown function u may not be the regularity of u but that of \( {\mathcal {L}}u\) where \({\mathcal {L}}\) is a linear operator (such as a PDE or a graph Laplacian). We show that the approximation of u obtained...

Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduce...

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