The next episode of our mostly Monte Carlo seminar is next Friday (9 October) at PariSanté Campus (room #8) with speakers
15:00 – Víctor Elvira, University of Edinburgh
16:00 – Edoardo Bandoni, Université Paris Dauphine-PSL
Víctor Elvira, “Rethinking self-normalized importance sampling”
Self-normalized importance sampling (SNIS) is one of the most widely used Monte Carlo techniques for inference with unnormalized target distributions. Despite its usefulness, SNIS is often viewed simply as a normalized version of ordinary importance sampling, and many of its methodological questions remain largely unexplored. In this talk, we revisit SNIS from a unified perspective. We first introduce a generalized formulation of self-normalized importance sampling based on coupled proposals, showing that the classical SNIS estimator is only one member of a broader family of Monte Carlo estimators with new opportunities for variance reduction. We then consider the classical SNIS estimator and present adaptive algorithms that learn proposals tailored to its optimal proposal distribution, together with theoretical guarantees including consistency, asymptotic normality, and convergence of the proposal. Together, these developments suggest that self-normalized importance sampling should be regarded as a distinct Monte Carlo methodology, with its own theory, optimality principles, and algorithmic design.
Edoardo Bandoni, “Rate-Optimal Randomised Kernel Quadrature”
Kernel quadrature is widely used to approximate integrals of smooth functions, with the worst-case error typically decaying at the minimax rate n-α/d for smoothness α in dimension d. Existing rate-optimal methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and less robust in practice. In this work, we study randomised quadrature methods with a focus on robustness rather than kernel-specific optimality. By minimising a tractable upper bound on the worst-case error, we obtain an explicit sampling distribution p*∝ πg with g=2d/(2α+d), which depends on the integration density π and on a but not on the kernel beyond its Sobolev order. Under a weak doubling condition on the design measure, independent samples from p* attain the minimax rate n-α/d. These assumptions cover a broad class of targets on compact and unbounded domains; we verify them explicitly for Beta-type densities, Gaussian measures, and Student-t distributions, the last of which yields the minimax rate n-min(α,(n+d/2)/d. This kernel-agnostic design improves robustness while maintaining optimal rates, and it applies beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal randomised quadrature.
Managed to 
Nicola Branchini presented his work with Victor Elvira (a close friend and coauthor, incidentally one of the organisers of the workshop!) on improving self normalised importance sampling by interpreting it as a ratio of estimators based on two samples (which may be the same) and attempting to optimise the joint distribution of said sample. The starting assumption is having (good) marginal importance functions, which means the goal here is in optimising a copula distribution targeting the ratio as quantity of interest. Optimality is however defined in terms of the approximate asymptotic variance of the ratio, which remains an approximation. The idea is nonetheless quite interesting and shows potential for connecting with bridge sampling and… 
Daniel Paulin presented a novel unbiased method based on kinetic Langevin dynamics that combines advanced splitting methods with enhanced gradients, avoiding Metropolis correction by coupling and multilevel Monte Carlo approach, achieving unbiasedness by telescoping, but involving an avalanche of acronyms in the leapfrog/Gibbs steps. And Adam Johansen (U of Warwick) on 
