# Black-box constructions for exchangeable sequences of random multisets

## TL;DR

This paper introduces generalized constructions for exchangeable sequences of point processes, specifically negative binomial processes with random base measures, broadening applicability in Bayesian nonparametrics and handling complex base measures.

## Contribution

It extends existing finitary constructions to any random base measure, enabling new models for negative binomial processes with complex, possibly infinite, base measures.

## Key findings

- Provides constructions for negative binomial processes with arbitrary random base measures.
- Enables modeling of complex multisets in Bayesian nonparametrics.
- Generalizes previous beta process-based constructions.

## Abstract

We develop constructions for exchangeable sequences of point processes that are rendered conditionally-i.i.d. negative binomial processes by a (possibly unknown) random measure called the base measure. Negative binomial processes are useful in Bayesian nonparametrics as models for random multisets, and in applications we are often interested in cases when the base measure itself is difficult to construct (for example when it has countably infinite support). While a finitary construction for an important case (corresponding to a beta process base measure) has appeared in the literature, our constructions generalize to any random base measure, requiring only an exchangeable sequence of Bernoulli processes rendered conditionally-i.i.d. by the same underlying random base measure. Because finitary constructions for such Bernoulli processes are known for several different classes of random base measures--including generalizations of the beta process and hierarchies thereof--our results immediately provide constructions for negative binomial processes with a random base measure from any member of these classes.