On Metrizing Vague Convergence of Random Measures with Applications on Bayesian Nonparametric Models

TL;DR

This paper establishes conditions for the vague convergence of certain random measures and applies these results to Bayesian nonparametric models, including deriving a finite approximation of the beta process and demonstrating its effectiveness through simulations.

Contribution

The paper provides a new characterization of vague convergence for random measures and applies it to develop a finite approximation of the beta process in Bayesian nonparametrics.

Findings

Vague convergence of measures is characterized by convergence of finite truncations.

Finite approximation of the beta process is derived from the main theoretical result.

Simulation shows the proposed method outperforms existing algorithms.

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 $\Pi$, $(p_{i,n})_{1 \le i \le n}$ are random variables chosen according to certain procedures and are independent of $(\theta_i)_{i \geq 1}$ and $\delta_{\theta_i}$ denotes the Dirac measure at $\theta_i$. We show that $\mu_{n}$ converges vaguely to $\mu=\sum_{i=1}^{\infty} p_{i} \delta_{\theta_i}$ if and only if $\mu^{(k)}_{n}=\sum_{i=1}^{k} p_{i,n} \delta_{\theta_i}$ converges vaguely to $\mu^{(k)}=\sum_{i=1}^{k} p_{i} \delta_{\theta_i}$ for all $k$ fixed. The limiting process $\mu$ 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.