相关论文: Symmetry Exploits for Bayesian Cubature Methods

This paper focusses on the formulation of numerical integration as an inferential task. To date, research effort has largely focussed on the development of Bayesian cubature, whose distributional output provides uncertainty quantification…

Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential…

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…

Automatic cubatures approximate multidimensional integrals to user-specified error tolerances. For high dimensional problems, it makes sense to fix the sampling density but determine the sample size, $n$, automatically. Bayesian cubature…

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and…

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and…

Gau{\ss} cubature (multidimensional numerical integration) rules are the natural generalisation of the 1D Gau{\ss} rules. They are optimal in the sense that they exactly integrate polynomials of as high a degree as possible for a particular…

A method is developed to compute analytically fully symmetric cubature rules on the triangle by using symmetric polynomials to express the two kinds of invariance inherent in these rules. Rules of degree up to 15, some of them new and of…

It is a widely held view that analytical integration is more accurate than the numerical one. In some special cases, however, numerical integration can be more advantageous than analytical integration. In our paper we show this benefit for…

73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than…

Integration over non-negative integrands is a central problem in machine learning (e.g. for model averaging, (hyper-)parameter marginalisation, and computing posterior predictive distributions). Bayesian Quadrature is a probabilistic…

Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with…

Bayesian quadrature (BQ) is a model-based numerical integration method that is able to increase sample efficiency by encoding and leveraging known structure of the integration task at hand. In this paper, we explore priors that encode…

We propose a novel approach for estimating conditional or parametric expectations in the setting where obtaining samples or evaluating integrands is costly. Through the framework of probabilistic numerical methods (such as Bayesian…

The sigma-point filters, such as the UKF, which exploit numerical quadrature to obtain an additional order of accuracy in the moment transformation step, are popular alternatives to the ubiquitous EKF. The classical quadrature rules used in…

We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the…

Automatic cubatures approximate integrals to user-specified error tolerances. For high dimensional problems, it is difficult to adaptively change the sampling pattern to focus on peaks because peaks can hide more easily in high dimensional…

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces…

Bayesian quadrature (BQ) is a sample-efficient probabilistic numerical method to solve integrals of expensive-to-evaluate black-box functions, yet so far,active BQ learning schemes focus merely on the integrand itself as information source,…

Gaussian graphical models are widely used to infer dependence structures. Bayesian methods are appealing to quantify uncertainty associated with structural learning, i.e., the plausibility of conditional independence statements given the…