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We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…
We present two different strategies to extend the classical multi-label chaining approach to handle imprecise probability estimates. These estimates use convex sets of distributions (or credal sets) in order to describe our uncertainty…
We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem…
Conformal prediction is a framework for providing prediction intervals with distribution-free validity, guaranteeing predictive coverage for data drawn from any distribution. Its two main variants are full conformal prediction and split…
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a…
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the…
We develop a domain-theoretic framework for imprecise probability reasoning and inference on general topological spaces with a countably based continuous lattice of open sets. We address two distinct forms of uncertainty: partial or…
Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density…
Dilation is a puzzling phenomenon within Imprecise Probability theory: when it obtains, our uncertainty evaluation on event $A$ is vaguer after conditioning $A$ on $B$, whatever is event $B$ in a given partition $\mathcal{B}$. In this paper…
Conformal prediction is a distribution-free technique for establishing valid prediction intervals. Although conventionally people conduct conformal prediction in the output space, this is not the only possibility. In this paper, we propose…
We show the existence of families of periodic polyhedra in spaces of constant curvature whose fundamental domains can be obtained by attaching prisms and antiprisms to Archimedean solids. These polyhedra have constant discrete curvature and…
The category of (abstract) fans is to the category of monoids what the category of schemes is to the category of rings: a fan is obtained by gluing spectra of monoids along open embeddings. Here we study the basic algebraic geometry of…
Effective uncertainty estimation is becoming increasingly attractive for enhancing the reliability of neural networks. This work presents a novel approach, termed Credal-Set Interval Neural Networks (CreINNs), for classification. CreINNs…
Conformal prediction (CP) produces prediction regions with finite-sample, distribution free coverage guarantees, but its interpretation as a quantitative uncertainty tool is often left implicit. We develop a category-theoretic approach that…
We describe the fundamental constructions and properties of determinantal probability measures and point processes, giving streamlined proofs. We illustrate these with some important examples. We pose several general questions and…
The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately…
Kernel density estimation is a popular method for estimating unseen probability distributions. However, the convergence of these classical estimators to the true density slows down in high dimensions. Moreover, they do not define meaningful…