The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers $\{1,\dots,n\}$. This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like $n^α$ with $α>0$. We prove a non-asymptotic concentration result for the number of parts of size $k=o(n^{α/(2α+4)}/(\log n)^{1/(2+α)})$. In particular, we show that these random variables concentrate around $c_{k}\,V_*\,n^α$ where $V_*\,n^α$ is the asymptotic number of parts and $c_k\approx k^{-(1+α)}$ is a positive value depending on $k$. We also obtain finite-$n$ bounds for the total number of parts. Our theorems complement asymptotic statements by Pitman and more recent results on large and moderate deviations by Favaro, Feng and Gao.
We study a Markov process constructed from the Pólya sum process, which yields a kind of spatial version of the Chinese restaurant process, where each 'table' is assigned a 'location'. This construction firstly allows a definition of summation of independent processes, and secondly a monotone coupling for different parameters. Moreover, the process is related to a multi-particle random walk on the positive integers and its hydrodynamic limit and fluctuations are discussed.
By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type $α$ (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors.
We construct a two-parameter family of Feller diffusions on the set of open subsets of $(0,1)$ that arise as diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains. The diffusions we construct are natural ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. Recently, there has been significant interest in ordered analogues of the diffusions Petrov constructed. Existing methods for constructing such processes have been based on pathwise methods using marked Lévy processes and an outstanding conj…
We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (2009). We relate such up-down CRPs to the splitting trees of Lambert (2010) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray-Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establi…
Recently there has been significant interest in constructing ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. One method for constructing these processes goes through taking an appropriate diffusive limit of Markov chains on integer compositions called ordered Chinese Restaurant Process up-down chains. The resulting processes are diffusions whose state space is the set of open subsets of the open unit interval. In this paper we begin to study nontrivial aspects of the order structure of these diffusions. In particular,…
Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-serve…
We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.
We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. At each time~$t$,~$R$ numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block's cardinality, independently. We prove results concerning the asymptot…
The purpose of this work is to describe a unified, and indeed simple, mechanism for non-parametric Bayesian analysis, construction and generative sampling of a large class of latent feature models which one can describe as generalized notions of Indian Buffet Processes(IBP). This is done via the Poisson Process Calculus as it now relates to latent feature models. The IBP was ingeniously devised by Griffiths and Ghahramani in (2005) and its generative scheme is cast in terms of customers entering sequentially an Indian Buffet restaurant and selecting previously sampled dishes as well as new dis…
We describe the combinatorial stochastic process underlying a sequence of conditionally independent Bernoulli processes with a shared beta process hazard measure. As shown by Thibaux and Jordan [TJ07], in the special case when the underlying beta process has a constant concentration function and a finite and nonatomic mean, the combinatorial structure is that of the Indian buffet process (IBP) introduced by Griffiths and Ghahramani [GG05]. By reinterpreting the beta process introduced by Hjort [Hjo90] as a measurable family of Dirichlet processes, we obtain a simple predictive rule for the gen…
