On functional central limit theorems of Bayesian nonparametric priors

A Bayesian nonparametric estimator to entropy is proposed. The derivation of the new estimator relies on using the Dirichlet process and adapting the well-known frequentist estimators of Vasicek (1976) and Ebrahimi, Pflughoeft and Soofi (1994). Several theoretical properties, such as consistency, of the proposed estimator are obtained. The quality of the proposed estimator has been investigated through several examples, in which it exhibits excellent performance.

show abstract

“…where c i is defined in (5). The next proposition underlines a direct connection between c i,a and c i .…”

Section: Bayesian Estimation Of the Entropymentioning

confidence: 79%

A Bayesian Nonparametric Estimation to Entropy

Al-Labadi

Patel

Vakiloroayaei

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…When P is Dirichlet process or the normalized inverse Gaussian process, the above conclusion has been known (e.g. Labadi and Zarepour, 2013;Labadi and Abdelrazeq, 2016).…”

Section: Remark 46 Assumption 41 Implies the Conditions (I) And (Ii) In Theorem 44mentioning

confidence: 94%

“…Labadi and Zarepour (2013) present the functional central limit theorem for the normalized inverse Gaussian process on D(R) when its parameter a is large by using its finite dimensional joint density. Labadi and Abdelrazeq (2016) obtain the functional central limit theorem for the Dirichlet process by using the finite dimensional densities and for the Beta process on D(R) by using the characteristic function.…”

Section: Introductionmentioning

confidence: 99%

Functional Central Limit Theorems for Stick-Breaking Priors

Zhang

2022

Bayesian Anal.

View full text Add to dashboard Cite

We obtain the strong law of large numbers, Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various Bayesian nonparametric priors which include the stick-breaking process with general stickbreaking weights, the two-parameter Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process. For the stick-breaking process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem and functional central limit theorem hold. Except in the case of the generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the method of moments to obtain the convergence results. For the generalized Dirichlet process we use its marginal distributions to obtain the asymptotics although the computations are highly technical.

show abstract

“…In Lemma 4, we have shown that the Lévy metric metrizes vague convergence in M (k) . We will use this to prove a similar result in M. As in Grandell (1977), the idea is to associate to each µ ∈ M a vector µ (1) , µ (2) , . .…”

Section: Let C +mentioning

confidence: 96%

“…This paper is organized as follows. Section 2 discusses metrizing the vague convergence of random measures of the form (1). It also develops a criterion for which µ n converges vaguely to µ.…”

Section: Introductionmentioning

confidence: 99%

On metrizing vague convergence of random measures with applications on Bayesian nonparametric models

Al-Labadi

2018

Statistics

View full text Add to dashboard Cite

This paper deals with studying vague convergence of random measures of the form µn = n i=1 pi,nδ θ i , where (θi) 1≤i≤n is a sequence of independent and identically distributed random variables with common distribution Π, (pi,n) 1≤i≤n are random variables chosen according to certain procedures and are independent of (θi) i≥1 and δ θ i denotes the Dirac measure at θi. We show that µn converges vaguely to µ = ∞ i=1 piδ θ i if and only if µ (k) n = k i=1 pi,nδ θ i converges vaguely to µ (k) = k i=1 piδ θ i for all k fixed. The limiting process µ plays a central role in many areas in statistics, including Bayesian nonparametric models. A finite approximation of the beta process is derived from the application of this result.A simulated example is incorporated, in which the proposed approach exhibits an excellent performance over several existing algorithms.

show abstract

On functional central limit theorems of Bayesian nonparametric priors

Cited by 10 publications

References 28 publications

A Bayesian Nonparametric Estimation to Entropy

A Bayesian Nonparametric Estimation to Entropy

Functional Central Limit Theorems for Stick-Breaking Priors

On metrizing vague convergence of random measures with applications on Bayesian nonparametric models

Contact Info

Product

Resources

About