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The concepts of independence and totalness of subspaces are introduced in the context of quasi-probability distributions in phase space, for quantum systems with finite-dimensional Hilbert space. It is shown that due to the…
Propensity scores are often used for stratification of treatment and control groups of subjects in observational data to remove confounding bias when estimating of causal effect of the treatment on an outcome in so-called potential outcome…
Simple function classes have emerged as toy problems to better understand in-context-learning in transformer-based architectures used for large language models. But previously proposed simple function classes like linear regression or…
Anthropic reasoning is a critical tool to understand probabilities, especially in a large universe or multiverse. According to anthropic reasoning, we should consider ourselves typical among members of a reference class that must include…
The paper introduces mixed networks, a new framework for expressing and reasoning with probabilistic and deterministic information. The framework combines belief networks with constraint networks, defining the semantics and graphical…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
We use Cramer's formula for the inverse of a matrix and a combinatorial expression for the determinant in terms of paths of an associated digraph (which can be traced back to Coates) to give a combinatorial interpretation of M\"obius…
We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent H\"{o}lder-continuous functions within rectangular domains. We establish a foundational approach to…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
The prevalence of spatially referenced multivariate data has impelled researchers to develop a procedure for the joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any…
We start by revisiting the derivation of the variational formulae for the functional assigning to a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains in certain…
It is the purpose of this article to compare various concepts of ``function spaces''. In particular we compare notions of the concept of Banach Function Spaces (in the spirit of Luxemburg-Zaanen) to the setting of solid BF-spaces as it is…
Ultrafunctions are a particular class of functions defined on a hyperreal field $\mathbb{R}^{\ast}\supset\mathbb{R}$. They have been introduced and studied in some previous works. In this paper we introduce a particular space of…
Bayesian inference in function space has gained attention due to its robustness against overparameterization in neural networks. However, approximating the infinite-dimensional function space introduces several challenges. In this work, we…
Functional data analysis is becoming increasingly popular to study data from real-valued random functions. Nevertheless, there is a lack of multiple testing procedures for such data. These are particularly important in factorial designs to…
Majorization theory is a powerful mathematical tool to compare the disorder in distributions, with wide-ranging applications in many fields including mathematics, physics, information theory, and economics. While majorization theory…
It is more and more frequently the case in applications that the data we observe come from one or more random variables taking values in an infinite dimensional space, e.g. curves. The need to have tools adapted to the nature of these data…
As animals interact with their environments, they must infer properties of their surroundings. Some animals, including humans, can represent uncertainty about those properties. But when, if ever, do they use probability distributions to…
Surprisal theory links human processing effort to the predictability of an upcoming linguistic unit, but empirical work often leaves the notion of a unit underspecified. In practice, experimental stimuli are segmented into linguistically…
Inferential relations govern our concept use. In order to understand a concept it has to be located in a space of implications. There are different kinds of conditions for statements, i.e. that the conditions represent different kinds of…