Loi multinomiale

Résumé

En théorie des probabilités, la loi multinomiale (aussi appelée distribution polynomiale) généralise la loi binomiale. Tandis que la loi binomiale concerne le nombre de succès lors d'une série de n épreuves de Bernoulli indépendantes, comme dans le jeu de pile ou face, la loi multinomiale ne se restreint pas aux épreuves comportant deux issues. La loi multinomiale s'applique par exemple au cas de n jets d'un dé à six faces : l'apparition du seul peut être modélisé par une loi binomiale alors que l'ensemble des apparitions des à 6 est modélisé par une loi multinomiale. Rappel sur la loi binomiale Loi binomiale vignette|gauche|redresse=.4|Pile ou face. On considère une pièce de monnaie où la probabilité d'obtenir « pile » est p. On considère une variable aléatoire binomiale X : il s'agit du nombre de « piles » obtenus pour n lancers d'une pièce de monnaie. La loi de probabilité s'écrit \mathbb{P}(X=k)

Source officielle

https://fr.wikipedia.org/wiki/Loi_multinomiale

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (8)

Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés (19)

Concepts associés

Chargement

Cours associés (19)

General graduate course on regression methods

Ce cours enseigne les notions élémentaires de la théorie de probabilité et de la statistique, tels que l'inférence, les tests et la régression.

The course gives an introduction to probability and statistics for physicists.

Afficher plus

Cours associés

Chargement

Séances de cours associées (32)

Séances de cours associées

Chargement

Olfactometer experiments are used to determine the effect of odours on the behaviour of organisms such as insects or nematodes, and typically result in data comprising many groups of small counts, overdispersed relative to the multinomial distribution. Overdispersion reflects a lack of independence or heterogeneity among individuals and can lead to statistics having larger variances than expected and possible losses of efficiency. In this thesis, some distributions which consist of generalisations of the multinomial distribution have been developed. These models are based on non-homogeneous Markov chain theory, take the overdispersion into account, and potentially provide a physical interpretation of the overdispersion seen in olfactometer data. Some inference aspects are considered, including comparison of the asymptotic relative efficiencies of three different sampling schemes. The fact that the empirical distributions well approximate the corresponding asymptotic distributions is checked. Observable differences in parameter estimates between data generated under different hypotheses are also studied. Finally, different models intended to shed light on various aspects of the data and/or the experiment procedure, are applied to three real olfactometer datasets.

EPFL2008

Chargement

Optimal Classification In Sparse Gaussian Graphic Model

Zhigang Yao

Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix Sigma, where the precision matrix Omega = Sigma(-1) is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of Omega, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when Omega is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where Omega is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.

Institute of Mathematical Statistics2013

Chargement

This paper presents a method for constructing confidence intervals using the jackknife. Asymptotic expansions are used to assess and reduce the skewness. It is proved that, although the error in the coverage rate remains of order O(1/n) for two-sided intervals, the main term due to the skewness in the distribution is eliminated, which constitute an improvement to the normal approximation that is important in the context of small samples. Several simulations are presented, with different distributions and functionals, and compared to the empirical accuracy obtained with a normal approximation. It is observed that our method produces in many cases a better empirical accuracy than the intervals due to a normal approximation, without requiring any knowledge on the underlying distribution.

1998

Chargement

Optimal Classification In Sparse Gaussian Graphic Model

Zhigang Yao

Institute of Mathematical Statistics2013