Independent finite approximations for Bayesian nonparametric inference

TL;DR

This paper introduces the automated independent finite approximation (AIFA), a new practical method for finite-dimensional approximation of Bayesian nonparametric models, providing error bounds and advantages over traditional truncated methods.

Contribution

The authors propose AIFA, a systematic and practical approach for finite approximations of CRMs and NCRMs, with error bounds and computational benefits, filling a gap in existing approximation techniques.

Findings

AIFA facilitates parallel inference and is easier to derive than TFAs.

Upper bounds on AIFA approximation error are established for common CRMs and NCRMs.

In real-data experiments, AIFA performs comparably to TFA and aids hyperparameter estimation.

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.