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

Luai Al-Labadi

Research output: Contribution to journalArticlepeer-review

Abstract

This paper deals with studying vague convergence of random measures of the form μn = ∑n i=1 pi,nδΘi, where (Θi)1≤i≤n is a sequence of independent and identically distributed random variables with common distribution Π, δΘi denotes the Dirac measure at Θi and pi,n are random variables, independent of Θii≥1, chosen according to certain procedures such that pi,n → pi almost surely, as n→∞, for fixed i. We show that, as n→∞, μn converges vaguely almost surely to μn = ∑ i=1 pi,nδΘi if and only if μn (k)n = ∑k i=1 pi,nδΘi converges vaguely almost surely to μ(k)n = ∑k i=1 pi,nδΘi 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.

Original languageEnglish
Pages (from-to)445-457
Number of pages13
JournalStatistics
Volume52
Issue number2
DOIs
Publication statusPublished - Mar 4 2018
Externally publishedYes

Keywords

  • Beta process
  • nonparametric Bayesian statistics
  • random measures
  • vague convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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