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13
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Regenerative composition structures
 ANN. PROBAB
, 2005
"... A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the po ..."
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Cited by 32 (18 self)
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A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].
The Bernoulli sieve
 Bernoulli
, 2004
"... Abstract. Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stickbreaking interval partition of [0 ..."
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Cited by 12 (3 self)
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Abstract. Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stickbreaking interval partition of [0, 1]. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts. 1. The Bernoulli sieve can be seen as a generalisation of the ‘game ’ found in [3]. The first round of the game starts with n players and amounts to tossing a coin with probability X1 for tails. Each of the players tosses one time and the players flipping tails must drop out. If all n get heads the trial is disqualified and must be repeated completely with all n players, as many times as necessary until some players do quit. If at least one player remains after the first round, the second round continues with the remaining players, who must toss another coin with probability X2 for tails. The game lasts with probabilities X3, X4,... for tails until all players are sorted out. It is assumed that the probabilities X1, X2,... are independent random variables with a given distribution ω on]0, 1 [ , and that given Xj the individual outcomes at
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordin ..."
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Cited by 11 (3 self)
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This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.
Hidden markov dirichlet process: Modeling genetic inference in open ancestral space
, 2007
"... We present a new statistical framework called hidden Markov Dirichlet process (HMDP) to jointly model the genetic recombinations among possibly infinite number of founders and the coalescencewithmutation events in the resulting genealogies. The HMDP posits that a haplotype of genetic markers is ge ..."
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Cited by 10 (4 self)
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We present a new statistical framework called hidden Markov Dirichlet process (HMDP) to jointly model the genetic recombinations among possibly infinite number of founders and the coalescencewithmutation events in the resulting genealogies. The HMDP posits that a haplotype of genetic markers is generated by a sequence of recombination events that select an ancestor for each locus from an unbounded set of founders according to a 1storder Markov transition process. Conjoining this process with a mutation model, our method accommodates both betweenlineage recombination and withinlineage sequence variations, and leads to a compact and natural interpretation of the population structure and inheritance process. An efficient sampling algorithm based on a twolevel nested Pólya urn scheme was also developed.
Prediction Rules for Exchangeable Sequences Related to Species Sampling
 IN PROCESSOR DESIGN. MASTER’S THESIS. LM ERICSSON 2000
, 1998
"... Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value wit ..."
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Cited by 9 (1 self)
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Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n , for j = 1; 2; : : :, and otherwise is a new value with distribution for some probability measure on S with no atoms. Then the pj;n depend only on the partitition of the first n integers induced by the first n values of the sequence. All possible distributions for such an exchangeable sequence are characterized in terms of constraints on the pj;n and in terms of their de Finetti representations.
Three sampling formulas
 Combin. Probab. Comput
, 2004
"... Abstract. Sampling formulas describe probability laws of exchangeable combinatorial structures like partitions and compositions. We give a brief account of two known parametric families of sampling formulas and add a new family to the list. 1 Introduction. By an integer composition of weight n and l ..."
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Cited by 8 (1 self)
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Abstract. Sampling formulas describe probability laws of exchangeable combinatorial structures like partitions and compositions. We give a brief account of two known parametric families of sampling formulas and add a new family to the list. 1 Introduction. By an integer composition of weight n and length ℓ we shall mean an ordered collection of positive integer parts λ = (λ1,...,λℓ); we write λ ⊢ n for ∑ λj = n. It will be convenient to also use variables Λk = λk +... + λℓ, k ≤ ℓ, so that λj = Λj − Λj−1. A composition structure is a nonnegative function q on compositions such that for each n the values {q(λ) : λ ⊢ n} comprise a probability distribution, say qn, and the qn’s satisfy
Bayesian nonparametric estimator derived from conditional Gibbs structures. Annals of Applied Probability
 J. Phys. A: Math. Gen
, 2008
"... We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predictin ..."
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Cited by 6 (2 self)
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We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required. 1. Introduction. Random
Spectrum: joint bayesian inference of population structure and recombination events
 Bioinformatics
, 2007
"... Motivation: While genetic properties such as linkage disequilibrium (LD) and population structure are closely related under a common inheritance process, the statistical methodologies developed so far mostly deal with LD analysis and structural inference separately, using specialized models that do ..."
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Cited by 4 (3 self)
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Motivation: While genetic properties such as linkage disequilibrium (LD) and population structure are closely related under a common inheritance process, the statistical methodologies developed so far mostly deal with LD analysis and structural inference separately, using specialized models that do not capture their statistical and genetic relationships. Also, most of these approaches ignore the inherent uncertainty in the genetic complexity of the data and rely on inflexible models built on a closed genetic space. These limitations may make it difficult to infer detailed and consistent structural information from rich genomic data such as populational SNP profiles. Results: We propose a new modelbased approach to address these issues through joint inference of population structure and recombination events under a nonparametric Bayesian framework; we present Spectrum, an efficient implementation based on our new model. We validated Spectrum on simulated data and applied it to two real SNP datasets, including singlepopulation Daly data and the fourpopulation HapMap data. Our method performs well relative to LDhat 2.0 in estimating the recombination rates and hotspots on these datasets. More interestingly, it generates an ancestral spectrum for representing population structures which not only displays substructure based on population founders but also reveals details of the genetic diversity of each individual. It offers an alternative view of the population structures to that offered by Structure 2.1, which ignores chromosomelevel mutation and combination with respect to founders. 1
On a Gibbs characterization of normalized generalized Gamma processes
, 707
"... We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion ..."
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Cited by 2 (0 self)
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We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type α ∈ (−∞,0].