Poisson-Kingman partitions

Abstract

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.

Exchangeable Gibbs partitions and Stirling triangles

“…connecting g α to the stable density f α , whose Laplace transform at λ is exp(−λ α ), we get V n,1 (s) (1 − α) n−1↑ = α Γ(1 − α)t α f α (t) 1 0 y n−1−α f α (t(1 − y))dy , which is an instance of [21,Equation (66)]. It follows that a path is regular if and only if κ ν /ν α converges, in which case the induced solution is ψ(α|s).…”

Posterior analysis of rare variants in Gibbs-type species sampling models

“…Under the framework (1), withP being in the class of the Gibbs-type random probability measures by Pitman [35], Lijoi et al [25] introduced a novel Bayesian nonparametric methodology for making inferences on quantities related to an additional unobserved sample (X n+1 , . .…”

“…Indeed, apart from the Dirichlet process, the class of the Gibbs-type random probability measures includes as special cases the two parameter Poisson-Dirichlet process, also known as Pitman-Yor process, and the normalized generalized Gamma process. We refer to Perman et al [33], Pitman and Yor [37] and Ishwaran and James [18] for details on the two parameter Poisson-Dirichlet process, and to James [19], Pitman [35], Lijoi et al [27] and James [20] for details on the normalized generalized Gamma process. Gibbs-type priors also stand out for being particularly suited in the context of inferential problems with a large unknown number of species, which typically occur in several genomic applications.…”

“…For α ∈ (−∞, 0) the extremes are Ewens-Pitman sampling models with parameter (α, −α ); for α = 0 the extremes are Ewens sampling models with parameter ≥ 0; for α ∈ (0, 1) the Ewens-Pitman sampling models are not extremes. We refer to Section 6.1 in Pitman [35] for details on W α . Example 1.…”

A note on Bayesian nonparametric priors derived from exponentially tilted Poisson–Kingman models

“…'s with inverse Gaussian distribution divided by their sum. Even if the authors observe that a N-IG prior, with non-atomic parameter measure, belongs to the class of species sampling models (Pitman, 1996), and more exactly to Poisson-Kingman models (Pitman, 2003), they do not derive such process as an element of the previous families, but obtain, independently of Pitman's results, both the analogous of the Blackwell-MacQueen prediction rules and the distribution of the number of distinct values observed in a sample.…”