A Conversation with Eugenio Regazzini

Eugenio Regazzini was born on August 12, 1946 in Cremona (Italy), and took his degree in 1969 at the University "L. Bocconi" of Milano. He has held positions at the universities of Torino, Bologna and Milano, and at the University "L. Bocconi" as assistant professor and lecturer from 1974 to 1980, and then professor since 1980. He is currently professor in probability and mathematical statistics at the University of Pavia. In the periods 1989-2001 and 2006-2009 he was head of the Institute for Applications of Mathematics and Computer Science of the Italian National Research Council (C.N.R.) in Milano and head of the Department of Mathematics at the University of Pavia, respectively. For twelve years between 1989 and 2006, he served as a member of the Scientific Board of the Italian Mathematical Union (U.M.I.). In 2007, he was elected Fellow of the IMS and, in 2001, Fellow of the "Istituto Lombardo---Accademia di Scienze e Lettere." His research activity in probability and statistics has covered a wide spectrum of topics, including finitely additive probabilities, foundations of the Bayesian paradigm, exchangeability and partial exchangeability, distribution of functionals of random probability measures, stochastic integration, history of probability and statistics. Overall, he has been one of the most authoritative developers of de Finetti's legacy. In the last five years, he has extended his scientific interests to probabilistic methods in mathematical physics; in particular, he has studied the asymptotic behavior of the solutions of equations, which are of interest for the kinetic theory of gases. The present interview was taken in occasion of his 65th birthday.Comment: Published in at http://dx.doi.org/10.1214/11-STS362 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Distributional properties of means of random probability measures

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.Bayesian Nonparametrics; Completely random measures; Cifarelli-Regazzini identity; Dirichlet process; Functionals of random probability measures; Generalized Stieltjes transform; Neutral to the right processes; Normalized random measures; Posterior distribution; Random means; Random probability measure; Two-parameter Poisson-Dirichlet process

Nonparametric priors for vectors of survival functions

The paper proposes a new nonparametric prior for two-dimensional vectors of survival functions (S1,S2). The definition we introduce is based on the notion of Lévy copula and it will be used to model, in a nonparametric Bayesian framework, two-sample survival data. Such an application will yield a natural extension of the more familiar neutral to the right process of Doksum (1974) adopted for drawing inferences on single survival functions. We, then, obtain a description of the posterior distribution of (S1,S2), conditionally on possibly right-censored data. As a by-product of our analysis, we find out that the marginal distribution of a pair of observations from the two samples coincides with the Marshall-Olkin or the Weibull distribution according to specific choices of the marginal Lévy measures.Bayesian nonparametrics; Completely random measures; Dependent stable processes; Lévy copulas; Posterior distribution; Right-censored data; Survival function

Distributional Properties of means of Random Probability Measures

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.Bayesian Nonparametrics; Completely random measures; Cifarelli–Regazzini identity; Dirichlet process; Functionals of random probability measures; Generalized Stieltjes transform; Neutral to the right processes; Normalized random measures; Posterior distribution; Random means; Random probability measure; Two–parameter Poisson–Dirichlet process.

Asymptotics for a Bayesian nonparametric estimator of species variety

In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently, it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper, we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two-parameter Poisson-Dirichlet process that was previously known only in the unconditional case.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ371 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Nonparametric Priors for Vectors of Survival Functions

The paper proposes a new nonparametric prior for two–dimensional vectors of survival functions (S1, S2). The definition we introduce is based on the notion of L´evy copula and it will be used to model, in a nonparametric Bayesian framework, two–sample survival data. Such an application will yield a natural extension of the more familiar neutral to the right process of Doksum (1974) adopted for drawing inferences on single survival functions. We, then, obtain a description of the posterior distribution of (S1, S2), conditionally on possibly right–censored data. As a by–product of our analysis, we find out that the marginal distribution of a pair of observations from the two samples coincides with the Marshall–Olkin or the Weibull distribution according to specific choices of the marginal L´evy measures.Bayesian nonparametrics, Completely random measures, Dependent stable processes, L´evy copulas, Posterior distribution, Right–censored data, Survival function

Models beyond the Dirichlet process

Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of exibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and ecient simulation techniques that make nonparametric models amenable to concrete use in a number of applied statistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process.

Bayesian inference with dependent normalized completely random measures

The proposal and study of dependent prior processes has been a major research focus in the recent Bayesian nonparametric literature. In this paper, we introduce a flexible class of dependent nonparametric priors, investigate their properties and derive a suitable sampling scheme which allows their concrete implementation. The proposed class is obtained by normalizing dependent completely random measures, where the dependence arises by virtue of a suitable construction of the Poisson random measures underlying the completely random measures. We first provide general distributional results for the whole class of dependent completely random measures and then we specialize them to two specific priors, which represent the natural candidates for concrete implementation due to their analytic tractability: the bivariate Dirichlet and normalized $\sigma$-stable processes. Our analytical results, and in particular the partially exchangeable partition probability function, form also the basis for the determination of a Markov Chain Monte Carlo algorithm for drawing posterior inferences, which reduces to the well-known Blackwell--MacQueen P\'{o}lya urn scheme in the univariate case. Such an algorithm can be used for density estimation and for analyzing the clustering structure of the data and is illustrated through a real two-sample dataset example.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ521 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm