Random Weighting, Asymptotic Counting, and Inverse Isoperimetry

For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of X and let Gamma(X,mu) be the expected maximum weight of a subset from X when the weights of 1,...,n are chosen independently at random from a symmetric probability distribution mu on R. We consider the inverse isoperimetric problem of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove that the optimal choice of mu is the logistic distribution, in which case Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X| grows. Since in many important cases Gamma(X,mu) can be easily computed, we obtain computationally efficient approximation algorithms for a variety of counting problems. Given mu, we describe families X of a given cardinality with the minimum value of Gamma(X,mu), thus extending and sharpening various isoperimetric inequalities in the Boolean cube.Comment: The revision contains a new isoperimetric theorem, some other improvements and extensions; 29 pages, 1 figur

Causal survival embeddings: non-parametric counterfactual inference under censoring

Model-free time-to-event regression under confounding presents challenges due to biases introduced by causal and censoring sampling mechanisms. This phenomenology poses problems for classical non-parametric estimators like Beran's or the k-nearest neighbours algorithm. In this study, we propose a natural framework that leverages the structure of reproducing kernel Hilbert spaces (RKHS) and, specifically, the concept of kernel mean embedding to address these limitations. Our framework has the potential to enable statistical counterfactual modeling, including counterfactual prediction and hypothesis testing, under right-censoring schemes. Through simulations and an application to the SPRINT trial, we demonstrate the practical effectiveness of our method, yielding coherent results when compared to parallel analyses in existing literature. We also provide a theoretical analysis of our estimator through an RKHS-valued empirical process. Our approach offers a novel tool for performing counterfactual survival estimation in observational studies with incomplete information. It can also be complemented by state-of-the-art algorithms based on semi-parametric and parametric models