The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks

2026-10-06

AI Summary

This study addresses the degeneration of predictive distributions in wide Bayesian neural networks, where mean-field variational inference becomes dominated by the prior. To mitigate this issue, the work introduces a likelihood tempering mechanism and investigates the evolution of predictive moments in single-hidden-layer linear networks under the infinite-width limit via asymptotic analysis, revealing phase transition behaviors of the mean and variance across different scaling regimes. The primary contributions include deriving exact predictive distributions in the infinite-width limit, establishing critical conditions for temperature decay that counteract degeneracy, and proposing a theoretical framework to recover Neural Network Gaussian Process (NNGP) posterior statistics. Collectively, these results provide rigorous theoretical foundations for reliable inference in high-dimensional Bayesian neural networks.

Abstract

In wide Bayesian neural networks, Gaussian mean-field variational inference is prone to "prior dominance": the Kullback-Leibler (KL) regularization term of the ELBO outweighs the expected log-likelihood, and the variational predictive distribution collapses to the prior predictive as the width $M$ grows. Tempering the likelihood, by raising it to the power $1/T$ for a temperature $T < 1$, is equivalent to scaling the KL term by $T$. We ask in this paper how fast $T$ must decrease with $M$ to counteract this degeneracy and strike a good balance between the two terms. For single-hidden-layer linear networks with isotropic Gaussian priors, we derive the limiting predictive distribution under schedules of the form $T = τ/M^{c}$, with constants $τ, c > 0$, as $M \to \infty$ and compare it with the untempered neural network Gaussian process (NNGP) posterior, the infinite-width limit of the exact posterior. Our main result is that the predictive expectation and variance undergo phase transitions at different scales: the limiting expectation leaves its prior value at $c = 1/2$, once $τ$ falls below an explicit threshold, and equals the least-squares prediction for $c > 1/2$, whereas the limiting variance keeps its prior value for $c < 1$, matches the NNGP's for $c=1$, and vanishes for $c > 1$. With suitable choices of $τ,c$, one can recover either the NNGP posterior expectation or its variance.

Problem

• Bayesian neural networks
• variational inference
• prior dominance
• likelihood tempering
• predictive moments

Innovation

• Likelihood Tempering
• Variational Bayesian Neural Networks
• Phase Transitions
• Neural Network Gaussian Process
• Prior Dominance