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 scale parameter of a Sobolev kernel (such as a Mat\' ern kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become``slowly"" overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model. The analysis is based on a combination of techniques from nonparametric regression and scattered data interpolation. Empirical results are provided in support of the theoretical findings.

Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many distinct weights as there are fully symmetric sets and that these weights can be solved from a linear system of equations constructed out of row sums of certain submatrices of the full kernel matrix. We present several numerical examples that show feasibility, both for a large number of nodes and in high dimensions, of the developed fully symmetric kernel quadrature rules. Most prominent of the fully symmetric kernel quadrature rules we propose are those that use sparse grids.

all of whom have been kind enough to host me at their home institutions, some of them several times, during the past four years. The probabilistic numerics research community is still compact enough for one to meet almost everybody during the short span of a few years. The various conferences, workshops, and visits have been made much more enjoyable and productive by the presence, often recurring, of in particular Dr.

The note is concerned with Gaussian filtering in non-linear continuous-discrete state-space models. We propose a novel Taylor moment expansion (TME) Gaussian filter which approximates the moments of the stochastic differential equation with a temporal Taylor expansion. Differently from classical linearisation or Itô-Taylor approaches, the Taylor expansion is formed for the moment functions directly and in time variable, not by using a Taylor expansion on the non-linear functions in the model. We analyse the theoretical properties, including the positive definiteness of the covariance estimate and stability of the TME filter. By numerical experiments, we demonstrate that the proposed TME Gaussian filter significantly outperforms the state-of-the-art methods in terms of estimation accuracy and numerical stability.

This paper focuses on the numerical computation of posterior expected quantities of interest, where existing approaches based on ergodic averages are gated by the asymptotic variance of the integrand. To address this challenge, a novel variance reduction technique is proposed, based on Sard's approach to numerical integration and the control functional method. The use of Sard's approach ensures that our control functionals are exact on all polynomials up to a fixed degree in the Bernstein-von-Mises limit, so that the reduced variance estimator approximates the behaviour of a polynomiallyexact (e.g. Gaussian) cubature method. The proposed estimator has reduced mean square error compared to its competitors, and is illustrated on several Bayesian inference examples. All methods used in this paper are available in the R package ZVCV.

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This article derives an accurate, explicit, and numerically stable approximation to the kernel quadrature weights in one dimension and on tensor product grids when the kernel and integration measure are Gaussian. The approximation is based on use of scaled Gauss-Hermite nodes and truncation of the Mercer eigendecomposition of the Gaussian kernel. Numerical evidence indicates that both the kernel quadrature and the approximate weights at these nodes are positive. An exponential rate of convergence for functions in the reproducing kernel Hilbert space induced by the Gaussian kernel is proved under an assumption on growth of the sum of absolute values of the approximate weights.

The sigma-point filters, such as the unscented Kalman filter, are popular alternatives to the ubiquitous extended Kalman filter. The classical quadrature rules used in the sigmapoint filters are motivated via polynomial approximation of the integrand; however, in the applied context, these assumptions cannot always be justified. As a result, a quadrature error can introduce bias into estimated moments, for which there is no compensatory mechanism in the classical sigma-point filters. This can lead in turn to estimates and predictions that are poorly calibrated. In this article, we investigate the Bayes-Sard quadrature method in the context of sigma-point filters, which enables uncertainty due to quadrature error to be formalized within a probabilistic model. Our first contribution is to derive the well-known classical quadratures as special cases of the Bayes-Sard quadrature method. Based on this, a general-purpose moment transform is developed and utilized in the design of a novel sigma-point filter, which explicitly accounts for the additional uncertainty due to quadrature error.