## Bernoulli

• Bernoulli
• Volume 25, Number 2 (2019), 1256-1288.

### 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.

#### Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1256-1288.

Dates
Revised: July 2017
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862850

Digital Object Identifier
doi:10.3150/18-BEJ1020

#### Citation

Campbell, Trevor; Huggins, Jonathan H.; How, Jonathan P.; Broderick, Tamara. Truncated random measures. Bernoulli 25 (2019), no. 2, 1256--1288. doi:10.3150/18-BEJ1020. https://projecteuclid.org/euclid.bj/1551862850

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#### Supplemental materials

• Supplement to “Truncated random measures”. Proofs for all results developed in this paper along with some additional example applications and simulations.