Abstract
A general approach to derive the weak convergence, when centered and rescaled, of certain Bayesian nonparametric priors is proposed. This method may be applied to a wide range of processes including, for instance, nondecreasing nonnegative pure jump Lévy processes and normalized nondecreasing nonnegative pure jump Lévy processes with known finite dimensional distributions. Examples clarifying this approach involve the beta process in latent feature models and the Dirichlet process.
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Al Labadi L, Zarepour M (2013a) A Bayesian nonparametric goodness of fit test for right censored data based on approximate samples from the beta-Stacy process. Can J Stat 41(3):466–487
Al Labadi L, Zarepour M (2013b) On asymptotic properties and almost sure approximation of the normalized inverse-Gaussian process. Bayesian Anal 8(3):553–568
Al Labadi L, Zarepour M (2014a) Goodness of fit tests based on the distance between the Dirichlet process and its base measure. J Nonparamet Stat 26(2):341–357
Al Labadi L, Zarepour M (2014b) On simulations from the two-parameter Poisson–Dirichlet process and the normalized inverse-Gaussian pn mrocess. Sankhyā A 76(1):158–176
Al Labadi L, Zarepour M (2015) Asymptotic properties of the beta process in latent feature models. arxiv:1411.3434
Antoniak CE (1974) Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann Stat 2(6):1152–1174
Bickel P, Freedman D (1981) Some asymptotic theory for the bootstrap. Ann Stat 9(6):1196–1217
Billingsley P (1999) Convergence of probability measures, 3rd edn. Wiley, New York
Brunner LJ, Lo AY (1996) Limiting posterior distributions under mixture of conjugate priors. Stat Sin 6(1):187–197
Conti PL (1999) Large sample Bayesian analysis for Geo/G/1 discrete-time queueing models. Ann Stat 27(6):1785–1807
Conti PL (2004) Approximated inference for the quantile function via Dirichlet processes. Metron 62(2):201–222
Dykstra RL, Laud P (1981) A Bayesian nonparametric approach to reliability. Ann Stat 9:35–367
Favaro S, Lijoi A, Nava C, Nipoti B, Prünster I, Teh YW (2016) On the stick-breaking representation for homogeneous NRMIs. Bayesian Anal 11(3):697–724
Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1(2):209–230
Ferguson TS, Klass MJ (1972) A Representation of independent increment processes without Gaussian components. Ann Math Stat 1:209–230
Härdle W, Müller M, Sperlich S, Werwatz A (2004) Nonparametric and semiparametric models. Springer, Berlin
Hjort NL (1990) Nonparametric Bayes estimators based on beta processes in models for life history data. Ann Stat 18(3):1259–1294
James LF (2008) Large sample asymptotics for the two-parameter Poisson–Dirichlet process. In: Clarke B, Ghosal S (ed) Pushing the limits of contemporary statistics contributions in Honor of Jayanta K. Ghosh, vol 3. IMS, Ohio, pp 187–199
James LF, Lijoi A, Prünster I (2006) Conjugacy as a distinctive feature of the Dirichlet process. Scand J Stat 33:105–120
James LF, Lijoi A, Prünster I (2009) Posterior analysis for normalized random measures with independent increments. Scand J Stat 36:7697
Kim Y, Lee J (2004) A Bernstein-von Mises theorem in the nonparametric rightcensoring model. Ann Stat 32(4):1492–1512
Kim N, Bickel P (2003) The limit distribution of a test statistic for bivariate normality. Stat Sin 13:327–349
Lijoi A, Mena RH, Prünster I (2005a) Hierarchical mixture modelling with normalized inverse Gaussian priors. J Am Stat Assoc 100(472):1278–1291
Lijoi A, Mena RH, Prünster I (2005b) Bayesian nonparametric analysis for a generalized Dirichlet process prior. Stat Inference Stoch Process 8(3):283–309
Lo AY (1983) Weak convergence for Dirichlet processes. Sankhyā A 45(1):105–111
Lo AY (1987) A large sample study of the Bayesian bootstrap. Ann Stat 15(1):360–375
Müller P, Quintana FA, Jara A, Hanson T (2015) Bayesian nonparametric data analysis. Springer, Switzerland
Phadia EG (2013) Prior processes and their applications: nonparametric Bayesian estimation. Springer, Berlin
Pitman J, Yor M (1997) The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann Probab 25(2):855–900
Pollard D (1984) Convergence of stochastic processes. Springer, New York
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Sethuraman J (1994) A constructive definition of Dirichlet priors. Stat Sin 4:639–650
Thibaux R, Jordan MI (2007) Hierarchical Beta processes and the Indian buffet process. In: Proceedings of the international conference on artificial intelligence and statistics (AISTATS)
van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes with applications to statistics. Springer, New York
Walker S, Muliere P (1997) Beta-stacy processes and a generalisation of the polya-urn scheme. Ann Stat 25(4):1762–1780
Zarepour M, Al Labadi L (2012) On a rapid simulation of the Dirichlet process. Stat Probab Lett 82(5):916–924
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Al Labadi, L., Abdelrazeq, I. On functional central limit theorems of Bayesian nonparametric priors. Stat Methods Appl 26, 215–229 (2017). https://doi.org/10.1007/s10260-016-0365-8
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DOI: https://doi.org/10.1007/s10260-016-0365-8
Keywords
- Beta process
- Dirichlet process
- Lévy processes
- Nonparametric Bayesian inference
- Processes with independent increments
- Quantile process
- Weak convergence