Exchangeable Gibbs partitions and Stirling triangles

By Alexander V. Gnedin, Jim Pitman. 2004

273 citations recorded by the source corpus.

AbstractFor two collections of nonnegative and suitably normalized weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaemOvay1aaSbaaSqaaiabd6gaUjabcYcaSiabdUga% RbqabaacbaGccqWFGaaicqWGxbWvdaWgaaWcbaWaaqWaaeaacqWGbb% qqdaWgaaadbaGaeGymaedabeaaaSGaay5bSlaawIa7aaqabaGccqWI% VlctcqWGxbWvdaWgaaWcbaWaaqWaaeaacqWGbbqqdaWgaaadbaGaem% 4AaSgabeaaaSGaay5bSlaawIa7aaqabaaaaa!507F! $$V_{n,k} W_{\left| {A_1 } \right|} \cdots W_{\left| {A_k } \right|} $$ , where |Aj| is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity. Bibliography: 29 titles.

Subfields: Applied Mathematics, Artificial Intelligence, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/04336ac4d81b9b65a52d146990253ea38f409969

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Citations in the corpus

Some exactly solvable models of urn process theory (2006, background): “Note finally, that Gnedin and Pitman have recently encountered similar generating functions in [45].”
Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams (2005, background): “A positive answer would lead to an alternative proof of Theorem 44 and could also be useful in studying Conjecture 45.”
Bayesian Nonparametric Estimation of the Probability of Discovering New Species (2007, methodology): “The notion of random exchangeable Gibbs partition is due to Pitman (2006) and is further considered, for different purposes, in recent papers by Gnedin & Pitman (2005), Berstycki & Pitman (2007) and in an International Center of Economic Research working paper by L.F. James, A. Lijoi and I.…”
Gibbs fragmentation trees (2007, methodology): “We start from a general Gibbs model (7) with w (1) = 1 and follow [7], Section 2 closely, where a similar characterization is derived in a partition rather than a tree context.”
Representations of the Infinite Symmetric Group (2016, methodology): “Examples can be found in Kerov [63, Chapter 1, Section 2], Gnedin–Pitman [50], Gnedin–Olshanski [46].”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, background): “the arXiv version [11] of [12] for a similar argument.”
Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “Indeed, if the cohesion function c( · ) in Hartigan’s definition depends on the cardinalities of the groups, a result of Gnedin and Pitman (2005a) states that it must be of the form c(nj) = (1− σ)nj−1 for j = 1, .”