A semi-parametric Bernstein-von Mises theorem at rate n−3/2 for frequency estimation
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 n3, 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.
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BibTeX
@misc{serra2026semiparametric,
title = {{A semi-parametric Bernstein-von Mises theorem at rate $n^-3/2$ for frequency estimation}},
author = {Paulo Serra},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00821},
eprint = {2611.00821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00821},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A semi-parametric Bernstein-von Mises theorem at rate n^−3/2 for frequency estimation
AU - Paulo Serra
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00821
UR - https://arxiv.org/abs/2611.00821
AB - 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.
ER -