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Bayesian nonparametric inference for discovery probabilities: credible intervals and large sample asymptotics

Julyan Arbel 1, 2 Stefano Favaro 2, 3 Bernardo Nipoti 2, 3 Yee Whye Teh 4
1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP [2007-2019] - Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019]
Abstract : Given a sample of size n from a population of individual belonging to different species with unknown proportions, a popular problem of practical interest consists in making inference on the probability Dn(l) that the (n+1)-th draw coincides with a species with frequency l in the sample, for any l=0,1,…,n. This paper contributes to the methodology of Bayesian nonparametric inference for Dn(l). Specifically, under the general framework of Gibbs-type priors we show how to derive credible intervals for the Bayesian nonparametric estimator of Dn(l), and we investigate the large n asymptotic behaviour of such an estimator. Of particular interest are special cases of our results obtained under the assumption of the two parameter Poisson-Dirichlet prior and the normalized generalized Gamma prior, which are two of the most commonly used Gibbs-type priors. With respect to these two prior assumptions, the proposed results are illustrated through a simulation study and a benchmark Expressed Sequence Tags dataset. To the best our knowledge, this illustration provides the first comparative study between the two parameter Poisson-Dirichlet prior and the normalized generalized Gamma prior in the context of Bayesian nonparemetric inference for Dn(l)
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Julyan Arbel, Stefano Favaro, Bernardo Nipoti, Yee Whye Teh. Bayesian nonparametric inference for discovery probabilities: credible intervals and large sample asymptotics. Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2017, 27, pp.839-858. ⟨10.5705/ss.202015.0250⟩. ⟨hal-01203324⟩

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