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

arXiv: https://arxiv.org/abs/2610.09132 · HTML: https://mlsanity.com/paper/2610.09132/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 47 references that are in the map.
- First posted on arXiv 6 Oct 2026.

## Builds on (most cited first, 3 of 3)

- [Exact posterior distributions of wide Bayesian neural networks](https://mlsanity.com/paper/2006.10541/) (2020)
- [Overpruning in Variational Bayesian Neural Networks](https://mlsanity.com/paper/1801.06230/) (2018)
- [Variational Inference of overparameterized Bayesian Neural Networks: a theoretical and empirical study](https://mlsanity.com/paper/2207.03859/) (2022)

## Abstract (arXiv)

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.