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

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 infinity 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.