Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.

Gaussian process regression underpins countless academic and industrial applications of machine learning and statistics, with maximum likelihood estimation routinely used to select appropriate parameters for the covariance kernel. However, it remains an open problem to establish the circumstances in which maximum likelihood estimation is well-posed, that is, when the predictions of the regression model are insensitive to small perturbations of the data. This article identifies scenarios where the maximum likelihood estimator fails to be well-posed, in that the predictive distributions are not Lipschitz in the data with respect to the Hellinger distance. These failure cases occur in the noiseless data setting, for any Gaussian process with a stationary covariance function whose lengthscale parameter is estimated using maximum likelihood. Although the failure of maximum likelihood estimation is part of Gaussian process folklore, these rigorous theoretical results appear to be the first of their kind. The implication of these negative results is that well-posedness may need to be assessed post-hoc, on a case-by-case basis, when maximum likelihood estimation is used to train a Gaussian process model.

Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson’s deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems—a setting for which no probabilistic numerical methods have yet been developed.

In this letter, we analyze certain student's t-filters for linear Gaussian systems with misspecified noise covariances. It is shown that under appropriate conditions, the filter both estimates the state and re-scales the noise covariance matrices in a Kullback-Leibler optimal fashion. If the noise covariances are misscaled by a common scalar, then the re-scaling is asymptotically exact. We also compare the student's t-filter scale estimates to the maximum-likelihood estimates. Simulations demonstrating the results on the Wiener velocity model are provided.

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 for the integral. However, the point estimators associated to Bayesian cubature can be inaccurate and acutely sensitive to the prior when the domain is high-dimensional. To address these drawbacks we introduce Bayes-Sard cubature, a probabilistic framework that combines the flexibility of Bayesian cubature with the robustness of classical cubatures which are well-established. This is achieved by considering a Gaussian process model for the integrand whose mean is a parametric regression model, with an improper prior on each regression coefficient. The features in the regression model consist of test functions which are guaranteed to be exactly integrated, with remaining degrees of freedom afforded to the non-parametric part. The asymptotic convergence of the Bayes-Sard cubature method is established and the theoretical results are numerically verified. In particular, we report two orders of magnitude reduction in error compared to Bayesian cubature in the context of a high-dimensional financial integral.

The aim of this article is to design a moment transformation for Student-t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable whose variance provides information about the incurred integration error. Advantage of the Student-t process quadrature over the traditional Gaussian process quadrature, is that the integral variance depends also on the function values, allowing for a more robust modelling of the integration error. The moment transform is applied in nonlinear sigma-point filtering and evaluated on two numerical examples, where it is shown to outperform the state-of-the-art moment transforms.