Bayes' theorem

Summary

In probability theory and statistics, Bayes' theorem (beɪz or beɪzɪz ; alternatively Bayes' law or Bayes' rule), and occasionally Bayes's theorem, named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of r

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https://en.wikipedia.org/wiki/Bayes'_theorem

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A l'heure où le canton de Neuchâtel ambitionne de n'être qu'un seul espace, le besoin de revaloriser Le Locle, notamment son centre inscrit au patrimoine mondial de l'Unesco et pourtant négligé, devient essentiel. En effet, de nombreuses friches, traces des années de crise que la ville a traversées, s'y sont généralisées et la ville peine à contenir ses habitants. Tandis que la municipalité présente des projets en marge de la ville, allant à l'encontre des préconisations de la LAT et du respect des valeurs paysagères, l'intervention propose au contraire d'exploiter le potentiel conséquent du tissu bâti existant. En effet, densifier permet non seulement de contenir l'étalement urbain, mais aussi de requalifier le centre-ville, qui devient alors articulation entre les deux versants. De fait, les traversées informelles rejoignent le maillage strict de l'urbanisme horloger et les grands axes routiers se font plus perméables. Le projet développe ainsi trois lieux stratégiques du tissu en générant des espaces connectés, identitaires et mixtes. Ces nouvelles polarités constituent des nœuds qui participent à la revitalisation de l'ensemble du centre-ville et favorisent les interactions sociales. Le parc de la Gare, l'ensemble Bournot et le square du quartier Neuf accueillent du logement et divers équipements publics, adaptés à la ville. Revisitant les typologies locloises, le projet amène une réflexion sur la manière d'habiter ensemble. Les logements contribuent ainsi à redonner envie de vivre au centre du Locle.

2016

We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large deviation estimates and the avalanche principle, effective.

2018

Byzantine Machine Learning Made Easy By Resilient Averaging of Momentums

Sadegh Farhadkhani, Rachid Guerraoui, Nirupam Gupta, Rafaël Benjamin Pinot, John Stephan

Byzantine resilience emerged as a prominent topic within the distributed machine learning community. Essentially, the goal is to enhance distributed optimization algorithms, such as distributed SGD, in a way that guarantees convergence despite the presence of some misbehaving (a.k.a., Byzantine) workers. Although a myriad of techniques addressing the problem have been proposed, the field arguably rests on fragile foundations. These techniques are hard to prove correct and rely on assumptions that are (a) quite unrealistic, i.e., often violated in practice, and (b) heterogeneous, i.e., making it difficult to compare approaches. We present RESAM (RESilient Averaging of Momentums), a unified framework that makes it simple to establish optimal Byzantine resilience, relying only on standard machine learning assumptions. Our framework is mainly composed of two operators: resilient averaging at the server and distributed momentum at the workers. We prove a general theorem stating the convergence of distributed SGD under RESAM. Interestingly, demonstrating and comparing the convergence of many existing techniques become direct corollaries of our theorem, without resorting to stringent assumptions. We also present an empirical evaluation of the practical relevance of RESAM.

PMLR2022