Abstract
Completely random measures (CRMs) and their normalizations (NCRMs) offer flexible models in Bayesian nonparametrics. But their infinite dimensionality presents challenges for inference. Two popular finite approximations are truncated finite approximations (TFAs) and independent finite approximations (IFAs). While the former have been well-studied, IFAs lack similarly general bounds on approximation error, and there has been no systematic comparison between the two options. In the present work, we propose a general recipe to construct practical finite-dimensional approximations for homogeneous CRMs and NCRMs, in the presence or absence of power laws. We call our construction the automated independent finite approximation (AIFA). Relative to TFAs, we show that AIFAs facilitate more straightforward derivations and use of parallel computing in approximate inference. We upper bound the approximation error of AIFAs for a wide class of common CRMs and NCRMs — and thereby develop guidelines for choosing the approximation level. Our lower bounds in key cases suggest that our upper bounds are tight. We prove that, for worst-case choices of observation likelihoods, TFAs are more efficient than AIFAs. Conversely, we find that in real-data experiments with standard likelihoods, AIFAs and TFAs perform similarly. Moreover, we demonstrate that AIFAs can be used for hyperparameter estimation even when other potential IFA options struggle or do not apply.
Funding Statement
Tin D. Nguyen, Jonathan Huggins, Lorenzo Masoero, and Tamara Broderick were supported in part by ONR grant N00014-17-1-2072, NSF grant CCF-2029016, ONR MURI grant N00014-11-1-0688, and a Google Faculty Research Award. Jonathan Huggins was also supported by the National Institute of General Medical Sciences of the National Institutes of Health under grant number R01GM144963 as part of the Joint NSF/NIGMS Mathematical Biology Program. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Citation
Tin D. Nguyen. Jonathan Huggins. Lorenzo Masoero. Lester Mackey. Tamara Broderick. "Independent Finite Approximations for Bayesian Nonparametric Inference." Bayesian Anal. 19 (4) 1187 - 1224, December 2024. https://doi.org/10.1214/23-BA1385
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