1,721,129 research outputs found
On a normalized random measure with independent increments relevant to Bayesian nonparametric inference
No full textRecently the class of normalized random measures with independent increments has been introduced. Such random probability measures, whose distributions act as nonparametric priors for Bayesian inference, are obtained by suitably normalizing time-changed increasing additive processes. We consider a specific normalized random measure with independent increments, which contains, as particular cases, the Dirichlet process, the normalized inverse Gaussian and stable random measures. Although its finite-dimensional distributions are not known, expressions for quantities of statistical interest can be derived. In particular, we provide simple rules for prior specification in terms of moments and obtain, in presence of exchangeable observations, its predictive distributions, which consist of a linear combination of the marginal distribution and of a weighted empirical distribution. We also study means of this random probability measure. Besides a necessary and suÿcient condition for finiteness, we derive the exact prior and posterior distribution of any mean
Misure di probabilità aleatorie derivate da processi additivi crescenti e loro applicazione alla statistica bayesiana
No full textRandom probability measures derived from increasing additive processes and their application in Bayesian Statistic
Random probability measures derived from increasing additive processes and their application to Bayesian statistics.
No full textIncreasing additive processes (IAP), i.e. processes with positive independent increments, represent a natural tool for defining random probability measures whose distributions act as nonparametric priors for Bayesian inference. It is well-known that the celebrated Dirichlet process can be obtained either by normalizing a time-changed gamma process or, as a particular case of neutral to the right (NTR) random probability measure, by the exponential transformation of a suitable IAP.
A new class of random probability measures is introduced by generalizing the former construction to any IAP: a normalized random measure with independent increments (RMI) is defined by a suitable normalization of a time-changed IAP. Even if their finite-dimensional distributions are not generally known, quantities of statistical interest turn out to have, for a large subclass, appealing forms leading to simple rules for prior specification and to predictive distributions which consist of a linear combination of the marginal distribution and of a weighted empirical distribution. Particular attention is devoted to the study of their means. Necessary and sufficient conditions for finiteness together with their exact prior and approximate posterior distributions are provided. Some illustrative examples of statistical relevance are considered in detail.
Normalized RMI can be further generalized to normalized IAP driven random measures, which contain the popular mixture of Dirichlet process as a particular case. Conditions for their existence are given. In particular, results for the distribution of means under both prior and posterior conditions are derived, and, relying upon the introduction of strategic latent variables, a full Bayesian analysis is undertaken.
Moreover, NTR random probability measures are considered. Their means can be represented as the exponential functional of an IAP. This fact is exploited in order to give sufficient conditions for finiteness of the mean and for absolute continuity of its distribution. In addition, expressions for its moments, of any order, are provided. By resorting to the maximum entropy algorithm, an approximation to the density of the mean of a NTR prior is obtained. The numerical results are compared to those yielded by some well-established simulation algorithms in the context of a survival analysis problem
Discussion of “On simulation and properties of the stable law” by L. Devroye and L. James
Full text linkDiscussion of “On simulation and properties of the stable law” by L. Devroye and L. Jame
Alcune considerazioni sulle elezioni presidenziali messicane del 2006
No full textAlcune considerazioni sulle elezioni presidenziali messicane del 200
Hierarchical mixture modeling with normalized inverse-Gaussian priors
No full textIn recent years the Dirichlet process prior has experienced a great success in the context of Bayesian mixture modeling. The idea of overcoming discreteness of its realizations by exploiting it in hierarchical models, combined with the development of suitable sampling techniques, represent one of the reasons of its popularity. In this article we propose the normalized inverse-Gaussian (N-IG) process as an alternative to the Dirichlet process to be used in Bayesian hierarchical models. The N-IG prior is constructed via its finite-dimensional distributions. This prior, although sharing the discreteness property of the Dirichlet prior, is characterized by a more elaborate and sensible clustering which makes use of all the information contained in the data. Whereas in the Dirichlet case the mass assigned to each observation depends solely on the number of times that it occurred, for the N-IG prior the weight of a single observation depends heavily on the whole number of ties in the sample. Moreover, expressions corresponding to relevant statistical quantities, such as a priori moments and the predictive distributions, are as tractable as those arising from the Dirichlet process. This implies that well-established sampling schemes can be easily extended to cover hierarchical models based on the N-IG process. The mixture of N-IG process and the mixture of Dirichlet process are compared using two examples involving mixtures of normals
Large sample properties of Gibbs-type priors
No full textIn this paper we concisely summarize some recent findings that can be found in De Blasi, Lijoi and Pruenster (2012) and concern large sample properties of Gibbs-type priors. We shall specifically focus on consistency according to the frequentist approach which postulates the existence of a “ true ” distribution P_0 that generates the data. We show that the asymptotic behaviour of the posterior is completely determined by the probability of obtaining a new distinct observation. Exploiting the predictive structure of Gibbs-type priors, we are able to establish that consistency holds essentially always for discrete P0 , whereas inconsistency may occur for diffuse P_0. Such findings are
further illustrated by means of three specific priors admitting closed form expressions and exhibiting a wide range of asymptotic behaviours
Discussion on the paper by Caron and Fox [Sparse graphs using exchangeable random measures/ Francois Caron and Emily Fox]
No full textDiscussion of "Sparse graphs using exchangeable random measures" by F. Caron and E. Fox in view of possible extensions to multi-sample contexts and testing
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