Last Layer Empirical Bayes

TL;DR

This paper introduces Last Layer Empirical Bayes (LLEB), a novel approach that uses a learnable prior via normalizing flow on the last layer of neural networks to improve uncertainty quantification.

Contribution

LLEB proposes a data-dependent prior modeled by a normalizing flow on the last layer, bridging Bayesian neural networks and ensembles for better uncertainty estimation.

Findings

LLEB performs on par with existing uncertainty quantification methods.

LLEB interpolates between BNNs and ensembles based on prior strength.

Empirical Bayes with normalizing flows is a promising research direction.

Abstract

The task of quantifying the inherent uncertainty associated with neural network predictions is a key challenge in artificial intelligence. Bayesian neural networks (BNNs) and deep ensembles are among the most prominent approaches to tackle this task. Both approaches produce predictions by computing an expectation of neural network outputs over some distribution on the corresponding weights; this distribution is given by the posterior in the case of BNNs, and by a mixture of point masses for ensembles. Inspired by recent work showing that the distribution used by ensembles can be understood as a posterior corresponding to a learned data-dependent prior, we propose last layer empirical Bayes (LLEB). LLEB instantiates a learnable prior as a normalizing flow, which is then trained to maximize the evidence lower bound; to retain tractability we use the flow only on the last layer. We show why LLEB is well motivated, and how it interpolates between standard BNNs and ensembles in terms of the strength of the prior that they use. LLEB performs on par with existing approaches, highlighting that empirical Bayes is a promising direction for future research in uncertainty quantification.