Approximating predictive probabilities of Gibbs-type priors

Julyan Arbel (1) , Stefano Favaro (2)

1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
2 Collegio Carlo Alberto

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 in terms of mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in a plethora of applications. Given a random sample of size n from an arbitrary Gibbs-type prior, we show that the corresponding predictive probabilities 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. Our result is illustrated through an extensive simulation study, which includes an application in the context of Bayesian nonparametric mixture modeling.

Mots clés: two parameter Poisson–Dirichlet prior; Gibbs-type prior; Bayesian nonparametrics; first and second order; predictive probabilities; normalized generalized Gamma prior; asymptotic approximations

Domaines: Calcul [stat.CO]; Théorie [stat.TH]; Applications [stat.AP]; Méthodologie [stat.ME]

Julyan Arbel, Stefano Favaro. Approximating predictive probabilities of Gibbs-type priors. Sankhya A, 2021, 83, pp.496-519. ⟨10.1007/s13171-019-00187-y⟩. ⟨hal-01693333⟩