Vectors of two-parameter Poisson–Dirichlet processes

Abstract

a b s t r a c tThe definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson-Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a σ -stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions.

Fully Nonparametric Regression for Bounded Data Using Dependent Bernstein Polynomials

“…Chung & Dunson, 2011). Dependent neutral to the right processes and correlated twoparameter Poisson-Dirichlet processes have been proposed by Epifani & Lijoi (2010) and Leisen & Lijoi (2011), respectively, by considering suitable Lévy copulas. The general class of dependent normalized completely random measures has been discussed, for instance, by Lijoi et al (2014).…”

A Dirichlet process functional approach to heteroscedastic-consistent covariance estimation

“…In principle, the MDP bootstrap can be extended to other (non-conjugate) Bayesian nonparametric of Gibbs-type (see Leisen & Lijoi, 2011;Bassetti, et al 2014;Zhu & Leisen, 2015;De Blasi et al, 2015). For each of these other prior processes, the variance V(n * | Z n , α) cannot be directly evaluated, because they do not provide explicit characterizations of the process variance.…”

Compound Random Measures and their Use in Bayesian Non-Parametrics

“…In this section, we shall define a general class of vectors of normalized random measures with independent increments that incorporates many recently proposed priors built by using normalization; see for instance Leisen and Lijoi (), Leisen et al . (), Zhu and Leisen (), Griffin et al .…”

“… where is the support of . Several specific constructions have been proposed including various forms of superposition (Griffin et al ., ; Lijoi and Nipoti, ; Lijoi et al ., ,b; Chen et al ., ; Bassetti et al ., ), kernel‐weighted completely random measures (Foti and Williamson, ; Griffin, ; Rosinski, ; Barndorff‐Nielsen et al ., ) and Lévy copula‐based approaches (Leisen and Lijoi, ; Leisen et al ., ; Zhu and Leisen, ). In this paper, we develop an alternative method for constructing correlated completely random measures which is tractable, whose properties can be derived and for which sampling methods for posterior inference without truncation can be developed.…”

“…The construction also provides a unifying framework for previously proposed constructions. Indeed, the σ ‐stable and gamma vector of dependent random measures that have been studied in the recent works of Leisen and Lijoi (), Leisen et al . () and Zhu and Leisen () are special cases.…”