Beta distribution

Summary

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution. The generali

Official source

https://en.wikipedia.org/wiki/Beta_distribution

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (100)

Related publications

Related people (6)

Related people

Related units (3)

Related units

Related concepts (66)

Related concepts

Related courses (55)

Related courses

Related lectures (97)

Related lectures

Embarquement immédiat. Un satellite clip-air pour Genève Aéroport

Guillaume Schobinger

On ne peut parler de ce projet sans parler de Clip-Air. C'est un avion modulaire constitué d'une aile porteuse sous laquelle sont "clipées" trois capsules. Ces dernières peuvent accueillir des passagers, du fret ou tout autre programme dépendant des besoins. La capsule, aux dimensions d'un wagon de train peut, une fois posée sur un wagon spécifique, poursuivre son trajet au-delà de l'aéroport en empruntant le réseau ferré. Le projet d'architecture consiste en l'élaboration d'un satellite de "clip" pour cet avion. Le site retenu est le tarmac de l'aéroport international de Genève. Implantée dans le prolongement de la première série de satellites planifiée en 1968, cette situation permet une connexion facile avec le reste de l'aéroport par les accès souterrains existants et permet également de rejoindre la gare CFF sans difficulté pour les capsules. Le projet organise un élément de distribution et d'orientation des capsules par transbordement parallèle. Au-dessous, est gérée l'arrivée des capsules, remplies préalablement dans des gares aménagées le long de la ligne ferroviaire ainsi que par l'arrivée des passagers en transit depuis le terminal principal de l'aéroport. Au-dessus se trouve l'ensemble des loges d'attente, shops, et restaurants, infrastructures vitales pour l'économie de l'aéroport. La structure du bâtiment évoque le dynamisme du vol. Une certaine légèreté et un équilibre dans la structure identifient le bâtiment au transport aérien. Posée sur six piles contenant l'ensemble des distributions verticales et les différents services, la structure en acier organise et construit l'espace.

2012

Deep neural networks have been empirically successful in a variety of tasks, however their theoretical understanding is still poor. In particular, modern deep neural networks have many more parameters than training data. Thus, in principle they should overfit the training samples and exhibit poor generalization to the complete data distribution. Counter intuitively however, they manage to achieve both high training accuracy and high testing accuracy. One can prove generalization using a validation set, however this can be difficult when training samples are limited and at the same time we do not obtain any information about why deep neural networks generalize well. Another approach is to estimate the complexity of the deep neural network. The hypothesis is that if a network with high training accuracy has high complexity it will have memorized the data, while if it has low complexity it will have learned generalizable patterns. In the first part of this thesis we explore Spectral Complexity, a measure of complexity that depends on combinations of norms of the weight matrices of the deep neural network. For a dataset that is difficult to classify, with no underlying model and/or no recurring pattern, for example one where the labels have been chosen randomly, spectral complexity has a large value, reflecting that the network needs to memorize the labels, and will not generalize well. Putting back the real labels, the spectral complexity becomes lower reflecting that some structure is present and the network has learned patterns that might generalize to unseen data. Spectral complexity results in vacuous estimates of the generalization error (the difference between the training and testing accuracy), and we show that it can lead to counterintuitive results when comparing the generalization error of different architectures. In the second part of the thesis we explore non-vacuous estimates of the generalization error. In Chapter 2 we analyze the case of PAC-Bayes where a posterior distribution over the weights of a deep neural network is learned using stochastic variational inference, and the generalization error is the KL divergence between this posterior and a prior distribution. We find that a common approximation where the posterior is constrained to be Gaussian with diagonal covariance, known as the mean-field approximation, limits significantly any gains in bound tightness. We find that, if we choose the prior mean to be the random network initialization, the generalization error estimate tightens significantly. In Chapter 3 we explore an existing approach to learning the prior mean, in PAC-Bayes, from the training set. Specifically, we explore differential privacy, which ensures that the training samples contribute only a limited amount of information to the prior, making it distribution and not training set dependent. In this way the prior should generalize well to unseen data (as it hasn't memorized individual samples) and at the same time any posterior distribution that is close to it in terms of the KL divergence will also exhibit good generalization.

EPFL2020

We study causality in gravitational systems beyond the classical limit. Using on-shell methods, we consider the 1-loop corrections from charged particles to the photon energy-momentum tensor - the self-stress - that controls the quantum interaction between two on-shell photons and one off-shell graviton. The self-stress determines in turn the phase shift and time delay in the scattering of photons against a spectator particle of any spin in the eikonal regime. We show that the sign of the beta-function associated to the running gauge coupling is related to the sign of time delay at small impact parameter. Our results show that, at first post-Minkowskian order, asymptotic causality, where the time delay experienced by any particle must be positive, is respected quantum mechanically. Contrasted with asymptotic causality, we explore a local notion of causality, where the time delay is longer than the one of gravitons, which is seemingly violated by quantum effects.

2022

EPFL2020