Bernstein-von Mises theorem for sparse generalized linear models
Abstract
We establish an oracle Bernstein-von Mises theorem for high-dimensional sparse generalized linear models under an outcome-independent spike-and-slab prior. The posterior consistently recovers the true active set and converges in total variation to the Gaussian law that would arise if this set were known in advance, even when the active dimension grows. The result holds for both ordinary and fractional posteriors, the latter with the expected variance inflation under tempering. In fixed-design logistic regression, a sufficient lower bound on the squared minimum signal is proportional to the logarithm of the ambient dimension divided by sample size. Separate likelihood controls for model selection and oracle approximation allow the Gaussian approximation to use a weaker curvature condition. We verify the conditions for six canonical and noncanonical models under fixed and random designs, including Poisson regression with potentially unbounded Fisher weights. The results also yield oracle-rate contraction, credible regions on the recovered support and pointwise frequentist coverage.
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BibTeX
@misc{li2026bernsteinvon,
title = {{Bernstein-von Mises theorem for sparse generalized linear models}},
author = {Hanqing Li and Xuewen Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.27137},
eprint = {2605.27137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27137},
}
RIS
TY - UNPB
TI - Bernstein-von Mises theorem for sparse generalized linear models
AU - Hanqing Li
AU - Xuewen Lu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2605.27137
UR - https://arxiv.org/abs/2605.27137
AB - We establish an oracle Bernstein-von Mises theorem for high-dimensional sparse generalized linear models under an outcome-independent spike-and-slab prior. The posterior consistently recovers the true active set and converges in total variation to the Gaussian law that would arise if this set were known in advance, even when the active dimension grows. The result holds for both ordinary and fractional posteriors, the latter with the expected variance inflation under tempering. In fixed-design logistic regression, a sufficient lower bound on the squared minimum signal is proportional to the logarithm of the ambient dimension divided by sample size. Separate likelihood controls for model selection and oracle approximation allow the Gaussian approximation to use a weaker curvature condition. We verify the conditions for six canonical and noncanonical models under fixed and random designs, including Poisson regression with potentially unbounded Fisher weights. The results also yield oracle-rate contraction, credible regions on the recovered support and pointwise frequentist coverage.
ER -