Poisson-Kingman partitions

By Jim Pitman. 2002

238 citations recorded by the source corpus.

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 two-parameter family of Poisson-Dirichlet 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.

Subfields: Artificial Intelligence, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/61b91cb8bf00a4ed09456acb5def0303772ab66a

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Citations in the corpus

Self-similar fragmentations derived from the stable tree I (2003, methodology): “Let us now discuss the conditioned forms of distributions of the sequence �T [0,x] given Tx. An easy way to obtain nice regular versions for these conditional laws is developed in [25, 27 ], and uses the notion of size-biased fragment.”
Combinatorial Stochastic Processes (2006, methodology): “[362] Check that the function defined by (4.”
Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics (2002, methodology): “Now letting Tn(p) := T − n(p) ∑ j=1 h(sj)(169) it is quite clear that one can gain further interpretation by adapting the descriptions given in Perman, Pitman and Yor (1992, section 4) and Pitman (1995b).”
Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages (2005, methodology): “This class has been discussed in varying generalities and contexts in, for instance, [18, 24, 35, 38, 39 , 40].”
Posterior Analysis for Normalized Random Measures with Independent Increments (2009, methodology): “In Pitman (1996, 2003) a thorough analysis of the two parameter Poisson-Dirichlet family, which can be generated by a stable subordinator, is provided.”
The Ubiquitous Ewens Sampling Formula (2016, background): “4 Gamma and stable subordinators For 0 ≤ α ≤ 1 and θ > 0, the theory of Poisson–Kingman partitions [75] brings forth a remarkable connection between Ewens’s sampling formula, the Poisson– Dirichlet distribution, and the jumps of subordinators.”
Stick-breaking autoregressive processes (2011, methodology): “2This process is also sometimes called a Pitman-Yor process, after Pitman and Yor (1997).”
Conjugacy as a Distinctive Feature of the Dirichlet Process (2006, background): “An allied class of random probability measures, the Poisson–Kingman models, independently proposed in a stimulating paper of Pitman (2003) not concerned with Bayesian applications, essentially coincides with the just defined normalized HRMIs under the additional assumption of diffuseness of (see…”
Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “See Pitman (2003) for a discussion on its representation as Poisson–Kingman model, a class of random distributions described in Section 3.3.”
Regenerative composition structures (2003, background): “We note in passing that a large family of symmetric composition structures was derived from the jumps of a subordinator in [35].”
Gibbs fragmentation trees (2007, background): “Kerov [11] showed that the only exchangeable partitions of N of Gibbs type of the two-parameter family PD( α,θ ) with usual range for parameters θ > − α , etc.; see also [7, 14].”
Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior (2005, methodology): “This follows immediately from Pitman (2003).”
Asymptotic laws for compositions derived from transformed subordinators (2004, methodology): “ts are known in the case when Re is derived by scaling the range of a subordinator (St,t∈[0,T]), that is, increasing process with stationary independent increments restricted to a ﬁnite time interval [1, 24, 25, 26]. (See [3] for general background on subordinators.) In this paper we study asymptoti”
Exchangeable Gibbs partitions and Stirling triangles (2004, methodology): “In [21] the (α|s)-partitions were derived from their Kingman’s representation, with the random closed set Z being the scaled range of an α-stable subordinator conditioned on its value at a fixed time.”
Self-similar fragmentations derived from the stable tree II: splitting at nodes (2004, methodology): “An easy way to obtain regular versions for these conditional laws is developed in [23, 24 ].”