Depth induces scale-averaging in overparameterized linear Bayesian neural networks

TL;DR

This paper explores how depth affects inference in overparameterized linear Bayesian neural networks, revealing that depth induces scale-averaging effects that influence the network's predictive behavior.

Contribution

It introduces a novel interpretation of finite deep linear Bayesian neural networks as data-dependent scale mixtures of Gaussian processes, unifying previous theoretical results.

Findings

Depth causes scale-averaging in predictions.

Finite networks differ from infinite-width limits.

Representation learning is influenced by depth.

Abstract

Inference in deep Bayesian neural networks is only fully understood in the infinite-width limit, where the posterior flexibility afforded by increased depth washes out and the posterior predictive collapses to a shallow Gaussian process. Here, we interpret finite deep linear Bayesian neural networks as data-dependent scale mixtures of Gaussian process predictors across output channels. We leverage this observation to study representation learning in these networks, allowing us to connect limiting results obtained in previous studies within a unified framework. In total, these results advance our analytical understanding of how depth affects inference in a simple class of Bayesian neural networks.