J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes42 Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....lies in the ith cycle of n . Associate with each cycle a random label taken independently from the distribution . Let Xn be the label of the cycle containing n 2 N. Then the sequence X is exchangeable and corresponds to the Dirichlet measure with parameter distribution . As it was shown in [K] [P2], the constructions leading to random Dirichlet measures admit natural generalizations. We now describe a generalized Chinese restaurant model ( K] The parameters of this model consist of a probability measure and a family fng of coherent central measures on the symmetric groups Sn . This means ....

J. Pitman, Some Developments of the Blackwell--MacQueen Urn Scheme, Statistics, Probability and Game theory (Ferguson T. S., Shapley L. S., MacQueen J. B. eds.), IMS Lecture Notes -- Monograph series, vol. 30, 1996, pp. 245--267.

....(j) ii) the bivariate sequence for (I (j) in a length biased order; iii) the bivariate sequence for (I (j) in an biased order; iv) the sequence of local times (L I j ) has the same distribution for (I j ) as j ) whereas the sequence of lengths ( I j ) does not. See [25, 26] for background about size biased random orderings. To illustrate the meaning of (iii) for instance, for each k 1, conditionally given the entire bivariate sequence ( I ) j=1;2; the probability of the event ) is L 1 , where L 1 = almost surely. And given also ....

....of a sequence (Q j ) known as the GEM( distribution after Griffiths, Engen and McCloskey. The distribution of (Q (j) obtained by ranking (Q j ) is known as the PoissonDirichlet distribution with parameter . See [17] 18, x9.6] 32] Lemma 15 (Characterizations of GEM( 23] 25] [26]) Fix 0 and 0. Let G and Q j ; j = 1; 2; be non negative random variables. Then the following are equivalent: i) the sequence (Q j ) admits the representation Q j = W j i=1 (1 Gamma W i ) where the W j are independent beta(1; variables, and G is independent of (Q j ) ....

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J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes42 Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

....and parking algorithms. This paper is a revision of the earlier preprint [42] See [48] for a broader context and further developments. 2 Preliminaries This section recalls some basic ideas from Kingman s theory of exchangeable random partitions [30, 31] as further developed in [43] See [45, 48] for more extensive reviews of these ideas and their applications. Except where otherwise specified, all random variables are assumed to be defined on some background probability space(# F , P) and E denotes expectation with respect to P. Let 2, let F denote a random probability ....

....k . P i k 1 . 2) The distribution of # n is determined by the distribution of the sequence of ranked frequencies (P i ) through the distribution of the size biased permutation ( P j ) To be precise, the EPPF p in (1) is given by the formula [43] P j # # . 3) Alternatively [45] (j 1 , j k ) j i (4) where (j 1 , j k ) ranges over all permutations of k positive integers, and the same formula holds with P j i replaced by P j i . For each n = 1, 2, the EPPF p, when restricted to (n 1 , n i = n, determines the distribution of # n . Since # ....

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J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

....together some facts about the Hermite function h , which is involved in the description of the random partition generated by Brownian excursions. 2 Preliminaries This section recalls some basic ideas from Kingman s theory of exchangeable random partitions, as further developed in [30] See [32] for a more extensive review of these ideas and their applications. Except where otherwise specified, all random variables are assumed to be defined on some background probability space( Omega ; F ; P) and E denotes expectation with respect to P. 2 Let ( P 1 ; P 2 ; Delta Delta ....

....to say that the lengths are i.i.d. gamma( b) variables. This model for random partitions has been extensively studied. It is well known that features of the pd( model can be derived by taking limits of this more elementary model with m i.i.d. gamma( b) lengths as 0 and m 1 with m . See [32] for a review of this circle of ideas and its applications to species sampling models. The pk(ae ff;C;b ) model for a random partition defined by ae ff;C;b in (46) for 0 ff 1 was proposed by McCloskey [26] who first exploited the key idea of size biased sampling in the setting of species ....

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J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture NotesMonograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

....with the beta(1; distribution P (V i 2 dx) 1 Gamma x) Gamma1 dx, so V i d = U 1= Gamma 1 for U with uniform distribution on [0; 1] The law of (P n ) defined by (5) for independent beta(1; variables V i , is commonly known as GEM( after Griffiths, Engen and McCloskey. See [8, 13, 21, 24] for background. Theorem 2 combined with Lemma 4 immediately yields the following result, part (i) of which is due to Gnedin Kerov [10, 11] for fi m = fi s = 1. Corollary 5 For j 0 let W j have the beta(1; j) distribution on [0; 1] Then (i) The (W j ; fi s ; fi m ) split and merge chain with ....

....of the random vector (P (N) n ) 1nk converges in total variation norm to that of (P n ) 1nk for (P n ) with GEM( distribution. 3 Size biased permutations In many applications of random discrete distributions, for instance to combinatorics [12] population genetics [8] species sampling [24], and models for coagulation and fragmentation [1, 5] the main feature of interest is the sizes of atoms of the distribution, rather than their labels or locations in some ambient space. For this reason, it is common to regard the state of some process of interest as an element of the set ....

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J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture NotesMonograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

.... as # # 0 reduces to a well known result for the Ewens sampling formula [44, 41.5) Easily from (99) and (96) for all (#, #) with 0 # 1 and # #, the P #,# distribution of #(# n ) n # converges as n ## to the distribution with density (#(# 1) #(# # 1) x # # g # (x) See also [66, 60] for further results related to this two parameter family of random partitions. 6 Distributions of some total variations In view of the simple exact formulae provided by Corollary 3 for the distributions of X (n) for X = B, B br , B ex , B me , it is natural to look for ....

J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

....discrete measure valued process, the ranked mass coalescent. Section 2. 3 presents a theorem which shows how certain operations of coagulation and fragmentation act on the two parameter family of distributions of exchangeable random partitions of N introduced in [31] and studied further in [33, 34]. Section 2.4 applies this theorem to the U coalescent to recover some of the results of Bolthausen Sznitman and to obtain various further developments. The conceptual framework of the paper is provided by Kingman s theory of exchangeable random partitions of N, as reviewed in Appendix A. 2 ....

.... proper, meaning P n f n = 1 almost surely, the EPF p is determined by the distribution P of f via the formula p(n 1 ; n k ) X (i 1 ; i k ) E 0 k Y j=1 f n j i j 1 A (67) where the sum is over all sequences of distinct positive integers (i 1 ; i k ) See also [31, 33] for simpler characterizations of the EPF in terms of the distribution of frequencies of blocks of Pi in order of their least elements. Acknowledgment Thanks to Steven Evans and the referee for a number of helpful suggestions, and to Vlada Limic and Warren Ewens for comments and corrections. ....

J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

....equal to some functions of Pi n and the collection of random sets ffi n : X i = ag : a an atom of g: 4) This can be established by a slight variation of the proof of Theorem 1 given in Section 2. The rest of this introduction shows how Theorem 1 combines with results obtained previously in [14] to yield a description of all possible functions p j;n and q n that could be used to generate an exchangeable sequence (Xn ) by a prediction rule of the form (3) for diffuse , and a corresponding description of the de Finetti representation of (Xn ) in terms of sampling from a random ....

....1; 2; and each partition fA 1 ; A k g of f1; ng, P( Pi n = fA 1 ; A k g) p(#A 1 ; #A k ) 5) for some non negative symmetric function p of finite sequences of positive integers n : n 1 ; n k ) Here #A is the number of elements of A. Following [12, 14], call p the exchangeable partition probability function (eppf) determined by Pi. Write k(n) for the length k of n : n 1 ; n k ) For each finite sequence n of positive integers and each 1 j k(n) 1, a finite sequence n j of positive integers is defined by incrementing n j by 1. ....

[Article contains additional citation context not shown here]

J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

.... a well known result for the Ewens sampling formula [43, 41.5) Easily from (99) and (96) for all (ff; with 0 ff 1 and Gammaff, the P ff; distribution of ( Pi n ) n ff converges as n 1 to the distribution with density ( Gamma( 1) Gamma( ff 1) x =ff g ff (x) See also [65, 59] for further results related to this two parameter family of random partitions. 6 Distributions of some total variations In view of the simple exact formulae provided by Corollary 3 for the distributions of jjX (n) jj for X = B;B br ; B ex ; B me , it is natural to look for corresponding ....

J. Pitman. Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson et al., editor, Statistics, Probability and Game Theory; Papers in honor of David Blackwell, volume 30 of Lecture Notes-Monograph Series, pages 245--267. Institute of Mathematical Statistics, Hayward, California, 1996.

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