Large deviation principles for the Ewens-Pitman sampling model

Favaro, Stefano; Feng, Shui

doi:10.1214/ejp.v20-3668

Cited by 7 publications

(6 citation statements)

References 34 publications

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“…x n e −xz 1/α f α (x)dx (9) displayed in Proposition 1 of Favaro et al [9]; the fifth follows from (5). To conclude, it is enough to show that the probability distribution of S α,θ G α θ+n,1 possesses a density coinciding with f (•; α, θ, n).…”

Section: A New Probabilistic Representation For K Nmentioning

confidence: 83%

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A Berry-Esseen theorem for Pitman's $α$-diversity

Dolera,

Favaro

2018

Preprint

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This paper is concerned with the study of the random variable Kn denoting the number of distinct elements in a random sample (X1, . . . , Xn) of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, PD(α, θ). For α ∈ (0, 1), Theorem 3.8 in [23] shows that Kn n α a.s.−→ S α,θ as n → +∞. Here, S α,θ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states thatholds with an explicit constant C(α, θ). The key ingredients of the proof are a novel probabilistic representation of Kn as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution. Finally, we present the following application in the context of Bayesian nonparametric inference for species sampling problems: given an initial (observable) random sample (X1, . . . , Xn) from the population, estimate of the number K (n) m of hitherto unseen species that would be observed in m additional (unobservable) samples. In the approach proposed in [8], (X1, . . . , Xn) is a random sample from PD(α, θ) featuring Kn = j ≤ n species (blocks), for which there holds K (n) m m α | (X1, . . . , Xn) a.s. −→ S α,θ (n, j) as m → +∞, where S α,θ (n, j) is related to S α,θ+n . Thus, we combine the previous main result with a new Berry-Esseen bound for de Finetti's theorem recently obtained in [6], to obtain another Berry-Esseen theorem for the convergence of the distribution of K (n) m m α | (X1, . . . , Xn).

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Section: A New Probabilistic Representation For K Nmentioning

confidence: 83%

“…e −nφz(y) f α (y)dy (11) for z > 0, where φ z (y) := zy − log y. This quantity is connected with (9) in view of the identity…”

Section: A Quantitative Laplace Methods For I N (Z)mentioning

confidence: 99%

A Berry-Esseen theorem for Pitman's $α$-diversity

Dolera,

Favaro

2018

Preprint

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“…This kind of Law of Large Numbers was first obtained by Pitman [14,Chapter 3] with no explicit convergence rates. Much more recently, Favaro, Feng and Gao [8,9] have used Pitman's explicit formulae to obtain large and moderate deviation results for the N n (k). Reference [9], which is the closest to our work, focuses on precise estimates for probabilities like (6)…”

Section: Resultsmentioning

confidence: 99%

“…In particular, the exact distribution of the random variables N n (k) we consider can be computed explicitly. Based on these formulae, [8], [9] obtained large and moderate deviation results for these variables. These results are briefly described in subsection 3.1 below.…”

Section: 2mentioning

confidence: 97%

Concentration in the Generalized Chinese Restaurant Process

Pereira¹,

Oliveira²,

Ribeiro³

2018

Preprint

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The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers {1, . . . , 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 ≈ 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.

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“…The fluctuation behaviour of K (n) m and M (n) l,m , as m goes to infinity, were studied in [6], where the notion of posterior α-diversity were introduced. Moreover, the associated large deviation principles have been recently established in [8] and [9].…”

Section: Introductionmentioning

confidence: 99%

Moderate Deviations for Ewens-Pitman Sampling Models

2018

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Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.

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Large deviation principles for the Ewens-Pitman sampling model

Cited by 7 publications

References 34 publications

A Berry-Esseen theorem for Pitman's $α$-diversity

A Berry-Esseen theorem for Pitman's $α$-diversity

Concentration in the Generalized Chinese Restaurant Process

Moderate Deviations for Ewens-Pitman Sampling Models

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