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

arXiv:2610.09132v1 Announce Type: new 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 $T 0$, 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 $\tau$ falls below an explicit threshold, and equals the least-squares prediction for $c > 1/2$, whereas the limiting variance keeps its prior value for $c 1$. With suitable choices of $\tau,c$, one can recover either the NNGP posterior expectation or its variance.