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Related papers: An information-theoretic upper bound on prime gaps

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There is great interest in predicting rare and extreme events in complex systems, and in particular, understanding the role of network topology in facilitating such events. In this work, we show that degree dispersion -- the fact that the…

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Research in epidemiology often focusses on designing interventions that result in the number of infected individuals asymptotically approaching zero, without considering that this number may peak at high values during transients. Recent…

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We propose a method based on finite mixture models for classifying a set of observations into number of different categories. In order to demonstrate the method, we show how the component densities for the mixture model can be derived by…

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Being probabilistic models, during inference large language models (LLMs) display rare events: behaviour that is far from typical but highly significant. By definition all rare events are hard to see, but the enormous scale of LLM usage…

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We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…

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Machine learning theory has mostly focused on generalization to samples from the same distribution as the training data. Whereas a better understanding of generalization beyond the training distribution where the observed distribution…

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I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain…

Number Theory · Mathematics 2010-10-12 Nilotpal Kanti Sinha

Mixture distributions are a workhorse model for multimodal data in information theory, signal processing, and machine learning. Yet even when each component density is simple, the differential entropy of the mixture is notoriously hard to…

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The concept of entropy rate for a dynamical process on a graph is introduced. We study diffusion processes where the node degrees are used as a local information by the random walkers. We describe analitically and numerically how the degree…

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Randomness in scientific estimation is generally assumed to arise from unmeasured or uncontrolled factors. However, when combining subjective probability estimates, heterogeneity stemming from people's cognitive or information diversity is…

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We study rare-event simulation for a class of problems where the target hitting sets of interest are defined via modern machine learning tools such as neural networks and random forests. This problem is motivated from fast emerging studies…

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Data Analysis, Statistics and Probability · Physics 2020-12-18 A. E. Allahverdyan , N. H. Martirosyan

We study rare events in networks with both internal and external noise, and develop a general formalism for analyzing rare events that combines pair-quenched techniques and large-deviation theory. The probability distribution, shape, and…

Physics and Society · Physics 2018-02-27 J. Hindes , I. B. Schwartz

The entropy is a measure of uncertainty that plays a central role in information theory. When the distribution of the data is unknown, an estimate of the entropy needs be obtained from the data sample itself. We propose a semi-parametric…

Methodology · Statistics 2022-01-06 Stéphane Robin , Luca Scrucca

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find…

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In this paper, we consider an importance sampling problem for a certain rare-event simulations involving the behavior of a diffusion process pertaining to a chain of distributed systems with random perturbations. We also assume that the…

Optimization and Control · Mathematics 2020-08-26 Getachew K. Befekadu

Random networks are a powerful tool in the analytical modeling of complex networks as they allow us to write approximate mathematical models for diverse properties and behaviors of networks. One notable shortcoming of these models is that…

Physics and Society · Physics 2023-07-10 Laurent Hébert-Dufresne , Márton Pósfai , Antoine Allard

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a…

General Mathematics · Mathematics 2020-02-19 Andrea Berdondini

Randomness in scientific estimation is generally assumed to arise from unmeasured or uncontrolled factors. However, when combining subjective probability estimates, heterogeneity stemming from people's cognitive or information diversity is…

Methodology · Statistics 2015-09-14 Ville A. Satopää , Robin Pemantle , Lyle H. Ungar