Abstract
The study of random partitions has been an active research area in probability over the last twenty years. A quantity that has attracted a lot of attention is the number of blocks in the random partition. Depending on the area of applications this quantity could represent the number of species in a sample from a population of individuals or he number of cycles in a random permutation, etc. In the context of Bayesian nonparametric inference such a quantity is associated with the exchangeable random partition induced by sampling from certain prior models, for instance the Dirichlet process and the two parameter Poisson-Dirichlet process. In this paper we generalize some existing asymptotic results from this prior setting to the so-called posterior, or conditional, setting. Specifically, given an initial sample from a two parameter Poisson-Dirichlet process, we establish conditional fluctuation limits and conditional large deviation principles for the number of blocks generated by a large additional sample.
Highlights
Among various definitions of the Ewens-Pitman sampling model, a simple and intuitive one arises from Zabell [27] in terms of the following urn model
As discussed in Lijoi et al [18], this statistic has direct applications in Bayesian nonparametric inference for species sampling problems arising from ecology, biology, genetics, linguistics, etc
Tm(n) = Rm (n) + Km(n), which is the number of blocks generated by the additional sample
Summary
Among various definitions of the Ewens-Pitman sampling model, a simple and intuitive one arises from Zabell [27] in terms of the following urn model. Asymptotics in a conditional Ewens-Pitman sampling model an additional ball of the same color with mass one. For α = 0 and θ > 0, Equation (1.4) is analogous to a law of large numbers type limit It was shown in Feng and Hoppe [10] that (log n)−1Kn satisfies a large deviation principle with speed log(n) and rate function of the form u log u θ. Xn+m), given Kn. As discussed in Lijoi et al [18], this statistic has direct applications in Bayesian nonparametric inference for species sampling problems arising from ecology, biology, genetics, linguistics, etc. In Bayesian nonparametric inference for species sampling problems, large m conditional asymptotic analysis are typically motivated by the need of approximating quantities of interest from the posterior distribution.
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