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31
Bayesian nonparametric inference for species variety with a twoparameter Poisson–Dirichlet process prior
, 2009
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Conditional formulae for Gibbstype exchangeable random partitions
 Ann. Appl. Probab
, 2013
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Record indices and ageordered frequencies in Exchangeable Gibbs Partitions
, 2008
"... Abstract We consider a random partition Π of N = {1, 2,...} such that, for each n, its restriction Πn to [n] = {1,..., n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n − αk)Vn+1,k + Vn+1,k+1 k ≤ ..."
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Cited by 7 (0 self)
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Abstract We consider a random partition Π of N = {1, 2,...} such that, for each n, its restriction Πn to [n] = {1,..., n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n − αk)Vn+1,k + Vn+1,k+1 k ≤ n = 1, 2,... (Gnedin and Pitman 2006). By ranking the blocks Πn1,..., Πnk of Πn by their ageorder i.e. by the order of their least elements i1,...,ik, we study how the distribution of the frequencies of the blocks depends on i1,...,ik. Several interesting representations for the limit ageordered relative frequencies X1, X2,... of Π arise, depending on which ij’s one conditions on. In particular, conditioning on the entire vector i = 1 = i1 < i2 <..., a representation is Xj = ξj−1 (1 − ξi) j = 1, 2,... i=j where the ξj’s are independent Beta random variables with parameters, respectively, (1−α, ij+1−αj−1). We show the connection of such a representation with the socalled BetaStacy class of random discrete distributions (Walker and Muliere 1997). The vector i is found to form a Markov chain depending on both α and V. When V is chosen from Pitman’s subfamily, the twoparameter GEM distribution is reobtained by averaging the ξ over i. Conditioning on ik alone, we give two alternative representations for the Laplace transform of both − log Xk and − log ( ∑k i=1 Xi), and we characterize Ewens ’ partitions as the only exchangeable Gibbs partitions for which − logXkik can be represented as an infinite sum of independent random variables. We finally show that, for every k, conditional on ∑k i=1 Xi, the distribution of the normalized ageordered frequencies X1 / ∑k i=1 Xi,..., Xk / ∑k i=1 Xi is a mixture of Dirichlet distributions on the (k − 1)dimensional simplex, whose mixing measure is indexed by ik. We provide a nontrivial explicit formula for the marginal distribution of ik. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens ’ partitions.
Regeneration in Random Combinatorial Structures
, 2009
"... Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we ..."
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Cited by 6 (2 self)
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Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the EwensPitman partitions, • studying asymptotic features of the regenerative compositions.
Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results. ∗†
, 805
"... By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type α (Gnedin and Pitman, 2006). Our con ..."
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Cited by 4 (3 self)
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By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type α (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors.
Inconsistency of Pitman–Yor Process Mixtures for the Number of Components
, 2014
"... In many applications, a finite mixture is a natural model, but it can be difficult to choose an appropriate number of components. To circumvent this choice, investigators are increasingly turning to Dirichlet process mixtures (DPMs), and Pitman–Yor process mixtures (PYMs), more generally. While thes ..."
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Cited by 4 (3 self)
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In many applications, a finite mixture is a natural model, but it can be difficult to choose an appropriate number of components. To circumvent this choice, investigators are increasingly turning to Dirichlet process mixtures (DPMs), and Pitman–Yor process mixtures (PYMs), more generally. While these models may be wellsuited for Bayesian density estimation, many investigators are using them for inferences about the number of components, by considering the posterior on the number of components represented in the observed data. We show that this posterior is not consistent—that is, on data from a finite mixture, it does not concentrate at the true number of components. This result applies to a large class of nonparametric mixtures, including DPMs and PYMs, over a wide variety of families of component distributions, including essentially all discrete families, as well as continuous exponential families satisfying mild regularity conditions (such as multivariate Gaussians).
Lookingbackward probabilities for Gibbstype exchangeable random partitions
, 2014
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