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.
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Cite this article
APA 7
Zhang, I., & Randrianarisoa, T. (2026). The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks. https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks
MLA 9
Zhang, Ian, and Thibault Randrianarisoa. "The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks." https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks.
Chicago (author–date)
Zhang, Ian, and Thibault Randrianarisoa. 2026. "The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks." https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks.
Harvard
Zhang, I. and Randrianarisoa, T. (2026) 'The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks', Available at: https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks.
Vancouver
Zhang I, Randrianarisoa T. The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks. https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks
IEEE
I. Zhang, and T. Randrianarisoa, "The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks," https://omanscience.com/en/articles/the-impact-of-likelihood-tempering-on-the-limiting-predictive-moments-of-variational-bayesian-linear-neural-networks.