optimal sampling for kernel quadrature on unbounded domains
My PhD student Edoardo Bandoni, along with Julien Stoehr and myself, completed a paper on validating (Bayesian) kernel quadrature with unbounded domains of integration. Which connects with probabilistic numerics, since the integrand is modelled as a Gaussian process. And RKHS methods. As opposed to Monte Carlo estimators, quadrature methods approximate integrals of smooth functions with worst-case error decaying at a minimax rate α/d for smoothness α in dimension d. Existing rate-optimal quadrature methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and thus less robust in practice. This paper studies instead randomised quadrature methods, with a focus on robustness rather than on kernel-specific optimality. We construct an explicit, n-dependent, sampling distribution that achieves minimax rates for worst-case errors over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design does improve robustness while retaining optimal rates and extends Briol et al. (2019) to the unbounded case. Which cannot always be easily handled by a change of variables. Our result thus mostly covers unbounded sampling measures such as Gaussian and Student-t distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal, randomised quadrature. [The above is mostly stated in the abstract.]
July 10, 2026 at 12:57 pm
Not only GP(0,k) is not a smoothing prior since it is not location-invariant, it assigns probability 0 to constant sign functions for, at least, any k going to 0 at infinity like Matérn and RBF.
How can we ever talk about (optimal) quadrature methods that exclude constant sign functions, especially for absolutely convergent Lebesgue integrals? I don’t know, you tell me.
Simply the worst “Bayesian” paper in quite some time, with the worst prior one can imagine. GP regression is Bayesian if and only if I’m the Pope. The authors are definitely not prepared for functional nonparametric Bayes. First, Bayes in finite dimension for a finite set of the function images, second Bayes in infinite dimension.
July 8, 2026 at 12:31 am
How is it possible that you don’t see that GP regression is anything but Bayesian and a monumental swindle?
If, ad absurdum, we have a likelihood p(X,Y|f), a prior GP(m,k) and GP regression is Bayesian, then by definition the joint posterior is
1/Z p(X,Y|f) GP(m,k)
which is supposed to be another GP whose moments involve the inverse of the nxn K=k(X,X) Gram matrix, n sample size, in the noiseless case (your paper, section 3).
It is impossible to get K^-1 from Bayes rule above: on the one hand, as is well known, computing K^-1 makes GP regression have O(n^3) generic computational complexity and we find tons of papers targeting sub-cubic complexity in special cases because that’s a serious drawback.
On the other hand, computing the joint posterior above has O(n) computational complexity as soon as p(X,Y|f) factorizes (e.g. i.i.d. Gaussian likelihood): just compute the n-1 products!
I can provide at least ten other proofs but one is enough.
I really don’t understand why so many people, including experts like you, don’t see that and lose their time with such junk maths, especially because truly O(n) Bayesian regression/interpolation is well-known (Wahba, Bretthorst, Ali-Djafari, Skilling, Idier, etc.)
Any idea please? Forgetting GP regression is the first step towards functional, nonparametric Bayes and applications.
July 3, 2026 at 12:06 pm
Between ghost GP priors that never appear in the equations and quadrature methods for absolutely convergent Lebesgue integrals that exclude almost everywhere positive functions from scratch, I fear the paper is mostly about optimal BS.