The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear 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/Mc, with constants τ, c > 0, as M → ∞ 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.
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BibTeX
@misc{zhang2026impact,
title = {{The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks}},
author = {Ian Zhang and Thibault Randrianarisoa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09132},
eprint = {2610.09132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09132},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks
AU - Ian Zhang
AU - Thibault Randrianarisoa
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09132
UR - https://arxiv.org/abs/2610.09132
AB - 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 → ∞ 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.
ER -