Paper:
A Bayes-Sard Cubature Method
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.
This paper argues that the perceived performance gap between probabilistic and non-probabilistic methods should be reconsidered. To this end, we consider a non-parametric Bayesian regression model augmented with a parametric component. The features in the parametric component, that is the pre-specified finite set of basis functions, will be denoted π. Then, an improper prior limit on the regression coefficients (see [48,33] and [42,Sec. 2.7]) is studied. This gives rise to Bayes-Sard cubature 1 (BSC), which differs at a fundamental level to standard BC, in that the functions in π are now exactly integrated. The extension is similar, though not identical, to that proposed in 1974 by Larkin [28] and in 1991 by O'Hagan [35], and non-probabilistic versions have appeared independently in [3,9] in the context of interpolation with conditionally positive definite kernels and optimal approximation in reproducing kernel Hilbert spaces. For other recent work, see also [40,Sec. 2.4]. Our contributions therefore include (i) establishing a coherent and comprehensive Gaussian process framework for BSC; (ii) rigorous study of convergence and conditions that need to be established on π; (iii) empirical experiments that demonstrate improved accuracy in high dimensions and robustness to misspecified kernel parameters compared to BC; and (iv) the important observation that, when the dimension of the function space π matches the number of cubature nodes, the BSC method can be used to endow any cubature rule with a meaningful probabilistic output.

Review:
The paper develops a new integration method which follows from a Gaussian process (GP) modeling of the integrand. The author starts by defining the posterior of the GP model, which appears as an approximation of the integrand. Fixing some parameters in the GP models, the author recovers the standard Bayesian cubature method (rk 2.3). The originality of their proposal, the Bayes-Sard cubature (BSC) method, comes from considering a limiting case in the GP model (which characterizes the most uninformative prior). The estimate given by the BSC method is the integral of the mean of the resulting posterior Gaussian process. It is a linear integration rule and another nice property is that it integrates perfectly the functions lying in some linear space pi (chosen by the user). The paper is reasonably organized and the ideas are well explained (with good examples). On top of that the paper surveys nicely many results by putting in light new links between some existing methods.   Here are some points/questions that might help to improve the paper: - The main contribution of the paper seems more methodological than theoretical but too few experiments are given in the end to validate the approach.  - The proposed method depends on a linear space pi (a notation which I find a bit exotic for a linear subspace). It is not clearly stated that the integrals of functions in pi should be known for the method to work. This being said, there might be some link with the control variates method as detailed for instance in the archived paper arXiv:1801.01797. In section 2.4 of the previous reference, the authors obtain a formulation with cubature weights of control variates (which integrates perfectly the functions in a linear space that correspond to your pi). In the present paper, the author gives some weights that, as well, are integrating perfectly the functions in pi. In Th 2.9 the weights are claimed to be unique. Does it means that both methods coincide? (the statement on uniqueness might require more details). - The statements of the theorems are too vague and it is sometimes difficult to understand their meaning. For instance, ‘’in the limit’’ is not clear while a standard equation would do the job. Especially, in Th 2.7, where one cannot know what means ‘’in the limit’’ without checking the proof. Another example is ‘’consider an n points cubature rule’’. This has not been defined. - Even if interesting, the convergence analysis is not the very point of the paper (as it follows from known results). It would be preferable to furnish some remarks on the rate and perhaps to compare with other methods (in place of theoretical consideration that would perfectly fit the supplementary material). - Section 2.1 results from classical techniques from the GP literature. It should be mentioned from the beginning (for the non expert reader). - page 3. line 105. As the meaning of the remark coincides with the definition of s and k (line 97), it seems to me redundant. - page 3. line 112. The remark should be at the beginning of section 2.1.3. - remark 2.13. It is said that the supplementary material ‘’elaborates’’ on assumptions (i) and (ii) whereas it just gives some definitions. I was disappointing to see no further explanation on the assumptions.