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

ABSTRACT

This paper deals with studying vague convergence of random measures of the form ${\mu }_n=\sum _{i=1}^n{p}_{i,n}{\delta }_{{\theta }_i}$, where $({\theta }_i{)}_{1\le i\le n}$ is a sequence of independent and identically distributed random variables with common distribution Π, ${\delta }_{{\theta }_i}$ denotes the Dirac measure at ${\theta }_i$ and $(p_{i,n}{)}_{1\le i\le n}$ are random variables, independent of $({\theta }_i{)}_{i\ge 1}$, chosen according to certain procedures such that $p_{i,n}\to p_i$ almost surely, as $n\to \mathrm{\infty }$, for fixed i. We show that, as $n\to \mathrm{\infty }$, ${\mu }_n$ converges vaguely almost surely to $\mu =\sum _{i=1}^{\mathrm{\infty }}{p}_i{\delta }_{{\theta }_i}$ if and only if ${\mu }_n^{\left(k\right)}=\sum _{i=1}^k{p}_{i,n}{\delta }_{{\theta }_i}$ converges vaguely almost surely to ${\mu }^{\left(k\right)}=\sum _{i=1}^k{p}_i{\delta }_{{\theta }_i}$ 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.

KEYWORDS:

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

MSC 2000:

• Primary: 62F15
• 60G57
• Secondary: 28A33

Acknowledgments

The author would like to offer his special thanks and appreciations to Jaeyong Lee for sharing his R codes.

Disclosure statement

No potential conflict of interest was reported by the author.