Asymptotics for a Bayesian nonparametric estimator of species variety

In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently, it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper, we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two-parameter Poisson-Dirichlet process that was previously known only in the unconditional case.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ371 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotics for a Bayesian nonparametric estimator of species richness

In Bayesian nonparametric inference, random discrete probability measures are commonly used as priors within hierarchical mixture models for density estimation and for inference on the clustering of the data. Recently it has been shown that they can also be exploited in species sampling problems: indeed they are natural tools for modeling the random proportions of species within a population thus allowing for inference on various quantities of statistical interest. For applications that involve large samples, the exact evaluation of the corresponding estimators becomes impracticable and, therefore, asymptotic approximations are sought. In the present paper we study the limiting behaviour of the number of new species to be observed from further sampling, conditional on observed data, assuming the observations are exchangeable and directed by a normalized generalized gamma process prior. Such an asymptotic study highlights a connection between the normalized generalized gamma process and the two–parameter Poisson–Dirichlet process that was previously known only in the unconditional case.Bayesian Nonparametrics; Species sampling models; Asymptotics; s–diversity; Polynomially and exponentially tilted random variables; Completely random measures; Normalized generalized gamma process; Two parameter Poisson–Dirichlet process.

Approximating predictive probabilities of Gibbs-type priors

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most "natural" generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in several applications. Given a sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same desirable features as the predictive probabilities of the two parameter Poisson-Dirichlet prior.Comment: 22 pages, 6 figures. Added posterior simulation study, corrected typo

A hybrid sampler for Poisson-Kingman mixture models

This paper concerns the introduction of a new Markov Chain Monte Carlo scheme for posterior sampling in Bayesian nonparametric mixture models with priors that belong to the general Poisson-Kingman class. We present a novel compact way of representing the infinite dimensional component of the model such that while explicitly representing this infinite component it has less memory and storage requirements than previous MCMC schemes. We describe comparative simulation results demonstrating the efficacy of the proposed MCMC algorithm against existing marginal and conditional MCMC samplers

Random variate generation and connected computational issues for the Poisson–Tweedie distribution

After providing a systematic outline of the stochastic genesis of the Poisson–Tweedie distribution, some computational issues are considered. More specifically, we introduce a closed form for the probability function, as well as its corresponding integral representation which may be useful for large argument values. Several algorithms for generating Poisson–Tweedie random variates are also suggested. Finally, count data connected to the citation profiles of two statistical journals are modeled and analyzed by means of the Poisson–Tweedie distribution

Random variate generation for exponential and gamma tilted stable distributions

We develop a new efficient simulation scheme for sampling two families of tilted stable distributions: exponential tilted stable (ETS) and gamma tilted stable (GTS) distributions. Our scheme is based on two-dimensional single rejection. For the ETS family, its complexity is uniformly bounded over all ranges of parameters. This new algorithm outperforms all existing schemes. In particular, it is more efficient than the well-known double rejection scheme, which is the only algorithm with uniformly bounded complexity that we can find in the current literature. Beside the ETS family, our scheme is also flexible to be further extended for generating the GTS family, which cannot easily be done by extending the double rejection scheme. Our algorithms are straightforward to implement, and numerical experiments and tests are conducted to demonstrate the accuracy and efficiency

Stochastic approximations to the Pitman-Yor process

In this paper we consider approximations to the popular Pitman-Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error epsilon goes to zero in terms of a polynomially tilted positive stable distribution. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the epsilon-version of the Pitman-Yor process.Comment: 19 page