Truncated random measures

Abstract

Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper we detail two major classes of sequential CRM representations-series representations and superposition representations-within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context.

Two part envelopes for rejection sampling of some completely random measures

“…The parameter σ has only a small effect on b opt (see Figure 1) and a simple choice is b = 4 5c , as for the beta process. Again, Campbell et al (2018) suggest an envelope which corresponds to b = 1.…”

“…The value of b opt as a function of c is shown in the first column of Figure 1, and b opt = 4 5c is an extremely good approximation. Campbell et al (2018) suggest an envelope which corresponds to b = 1.…”

“…The value of σ has a small effect on the value of b opt and I suggest using b opt = 0.8065. Campbell et al (2018) suggest an envelope which corresponds to b = ∞.…”

“…The number of rejected points, (Campbell et al, 2018). This integral will become smaller as φ(x) becomes closer to ν(x).…”

“…which was termed the Lomax process by Campbell et al (2018). The natural two-piece jump intensity is…”