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Exploiting the geometric nature of statistical divergences, we devise a way to define associated induced uncertainty measures for discrete and finite probability distributions. We also report new uncertainty measures and discuss their…
The problem of probabilistic verification of a neural network investigates the probability of satisfying the safe constraints in the output space when the input is given by a probability distribution. It is significant to answer this…
As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure,…
When computing quantum-mechanical observables, the ``curse of dimensionality'' limits the naive approach that uses the quantum-mechanical wavefunction. The semiclassical Herman--Kluk propagator mitigates this curse by employing a grid-free…
We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case…
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation,…
We study robust estimators of the mean of a probability measure $P$, called robust empirical mean estimators. This elementary construction is then used to revisit a problem of aggregation and a problem of estimator selection, extending…
A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical…
Quasi-Monte Carlo cubature methods often sample the integrand using Sobol' (or other digital) sequences to obtain higher accuracy than IID sampling. An important question is how to conservatively estimate the error of a digital sequence…
The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be…
Probability measures by themselves, are known to be inappropriate for modeling the dynamics of plain belief and their excessively strong measurability constraints make them unsuitable for some representational tasks, e.g. in the context of…
We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters.…
We consider the problem of classification of points sampled from an unknown probability measure on a Euclidean space. We study the question of querying the class label at a very small number of judiciously chosen points so as to be able to…
We introduce a sharpness functional for probabilistic models that quantifies sharpness as an intrinsic property of the probability distribution. The measure is derived based on a rank-based concentration principle that tracks upward…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the…
In number theory, great efforts have been undertaken to study the Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups. On the other hand, group theorists have studied a probability measure on the set of all…
Below is a method for relating a mixed volume computation for polytopes sharing many facet directions to a symmetric random walk. The example of permutahedra and particularly hypersimplices is expanded upon.
Let $G\curvearrowright M$ be an isometric action of a Lie Group on a complete orientable Riemannian manifold. We disintegrate absolutely continuous measures with respect to the volume measure of $M$ along the principal orbits of…
Pseudo-marginal Metropolis-Hastings (pmMH) is a versatile algorithm for sampling from target distributions which are not easy to evaluate point-wise. However, pmMH requires good proposal distributions to sample efficiently from the target,…