Related papers: Existence and Uniqueness of Proper Scoring Rules
In this paper, we extend the concept of generalized entropy to uniform spaces, allowing computations beyond metrizable settings. We apply this to parabolic dynamics - systems with a unique fixed point uniformly attracting all compact…
Deformations of geometric characteristics of statistical hypersurfaces governed by the law of growth of entropy are studied. Both general and special cases of deformations are considered. The basic structure of the statistical hypersurface…
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The…
We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate…
Shannon and Khinchin showed that assuming four information theoretic axioms the entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here we note that in physical systems one of these axioms may be violated. For non-ergodic…
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and,…
Estimating the entropy of a discrete random variable is a fundamental problem in information theory and related fields. This problem has many applications in various domains, including machine learning, statistics and data compression. Over…
The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and…
We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of…
A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by…
For a vector $v=(v_1,\dots ,v_n)$ with $v_1>\cdots>v_n$ and $\sum v_i=0$, we study the "directional entropy" of two arithmetic objects: (1) the logarithmic embeddings of degree-$n$ totally real units, and (2) the logarithmic eigenvalue data…
The essence of the second law of classical thermodynamics is the `entropy principle' which asserts the existence of an additive and extensive entropy function, S, that is defined for all equilibrium states of thermodynamic systems and whose…
Probabilistic programs are typically normal-looking programs describing posterior probability distributions. They intrinsically code up randomized algorithms and have long been at the heart of modern machine learning and approximate…
This paper concerns algorithms that give correct answers with (asymptotic) density $1$. A dense description of a function $g : \omega \to \omega$ is a partial function $f$ on $\omega$ such that $\left\{n : f(n) = g(n)\right\}$ has density…
We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the…
Given n (discrete or continuous) random variables X_i, the (2^n-1)-dimensional vector obtained by evaluating the joint entropy of all non-empty subsets of {X_1,...,X_n} is called an entropic vector. Determining the region of entropic…
Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of…
We describe an approach to improving model fitting and model generalization that considers the entropy of distributions of modelling residuals. We use simple simulations to demonstrate the observational signatures of overfitting on ordered…
We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting…
This paper develops a uniformly valid and asymptotically nonconservative test based on projection for a class of shape restrictions. The key insight we exploit is that these restrictions form convex cones, a simple and yet elegant structure…