{
  "abstract": "We consider Bayesian inference for the frequency of a periodic signal of unknown shape, observed in Gaussian white noise on the interval $[0,n]$. The information about the frequency grows like $n^3$, so the frequency can be estimated at rate $n^{-3/2}$, while the shape is an infinite-dimensional nuisance parameter. We endow the frequency and the shape with independent priors, the prior on the shape being a random series in the trigonometric basis. We give conditions on the priors under which the posterior for the signal contracts at a rate that, for suitable choices, is the minimax rate for Sobolev classes of smoothness at least one, up to a logarithmic factor and adaptively to the smoothness of the shape. If, moreover, the prior keeps the shape away from shapes with a shorter period, also the marginal posteriors for the frequency and the shape contract around the truth. Under further conditions, which hold for a Gaussian random series prior conditioned to keep the shape away from shapes with a shorter period, for every true shape with smallest period 1 and Sobolev smoothness larger than $3/2$, the marginal posterior for the frequency satisfies a semi-parametric Bernstein-von Mises theorem at rate $n^{-3/2}$, which we prove by moving the origin of time, a change that leaves the posterior unchanged when the prior on the shape is invariant under time shifts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00821",
  "author_total": 1,
  "authors": [
    {
      "name": "Paulo Serra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hw87rj0mzwvf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00821",
  "fulltext_url": null,
  "id": "0hw87rj0mzwvf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00821",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hw87rj0mzwvf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A semi-parametric Bernstein-von Mises theorem at rate $n^{-3/2}$ for frequency estimation",
  "title_text": "A semi-parametric Bernstein-von Mises theorem at rate n^−3/2 for frequency estimation",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/0hw87rj0mzwvf",
  "venue": "arXiv",
  "versions": []
}