Depth induces scale-averaging in overparameterized linear Bayesian neural networks
This work addresses a theoretical gap in Bayesian deep learning for researchers, but it is incremental as it builds on prior studies in a simplified model.
The paper tackled the problem of understanding inference in deep Bayesian neural networks by interpreting finite deep linear Bayesian neural networks as data-dependent scale mixtures of Gaussian process predictors, connecting previous limiting results within a unified framework to advance analytical understanding of depth effects.
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.