31 research outputs found
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