{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09132",
  "author_total": 2,
  "authors": [
    {
      "name": "Ian Zhang"
    },
    {
      "name": "Thibault Randrianarisoa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0g548srrnmkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09132",
  "fulltext_url": null,
  "id": "c0g548srrnmkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09132",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7220951483",
  "pdf_url": "https://arxiv.org/pdf/2610.09132",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0g548srrnmkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:48Z",
      "native_id": "2610.09132",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09132"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks",
  "title_text": "The Impact of Likelihood Tempering on the Limiting Predictive Moments of Variational Bayesian Linear Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:48Z",
  "url": "https://research.draconitic.com/paper/c0g548srrnmkv",
  "venue": "arXiv",
  "versions": []
}