Black-box constructions for exchangeable sequences of random multisets
This work provides a theoretical tool for Bayesian nonparametrics, specifically for modeling random multisets, but it is incremental as it builds on existing finitary constructions for Bernoulli processes.
The paper tackles the problem of constructing exchangeable sequences of negative binomial processes with random base measures, which are difficult to build in cases like infinite support, by generalizing from known Bernoulli process constructions to handle any random base measure, enabling immediate applications to classes like beta processes and their hierarchies.
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.