Accuracy of Gaussian approximation for high-dimensional posterior distributions
Spokoiny, Vladimir Panov, Maxim
Spokoiny, Vladimir
Panov, Maxim
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Department
Machine Learning
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Journal article
Date
2025
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Language
English
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Abstract
The prominent Bernstein – von Mises (BvM) result claims that the posterior distribution after centering by the efficient estimator and standardizing by the square root of the total Fisher information is nearly standard normal. In particular, the prior completely washes out from the asymptotic posterior distribution. This fact is fundamental and justifies the Bayes approach from the frequentist viewpoint. In the nonparametric setup the situation changes dramatically and the impact of prior becomes essential even for the contraction of the posterior; see (Ann. Statist. 36 (2008) 1435–1463, Ann. Statist. 39 (2011) 2557–2584, Ann. Statist. 41 (2013) 1999–2028, Ann. Statist. 42 (2014) 1941–1969) for different models like Gaussian regression or i.i.d. model in different weak topologies. This paper offers another non-asymptotic approach to studying the behavior of the posterior for a special but rather popular and useful class of statistical models and for Gaussian priors. Our main results describe the accuracy of Gaussian approximation of the posterior. In particular, we show that restricting to the class of all centrally symmetric credible sets around the penalized maximum likelihood estimator (pMLE) allows to get Gaussian approximation up to order
. We also derive tight finite sample bounds on posterior contraction in terms of the so-called effective dimension of the parameter space and address the question of frequentist reliability of Bayesian credible sets. The obtained results are specified for nonparametric log-density estimation and generalized regression.
Citation
V. Spokoiny and M. Panov, “Accuracy of Gaussian approximation for high-dimensional posterior distributions,” https://doi.org/10.3150/21-BEJ1412, vol. 31, no. 2, pp. 843–867, May 2025, doi: 10.3150/21-BEJ1412.
Source
Bernoulli
Conference
Keywords
Concentration, contraction Gaussian approximation, Posterior
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Bernoulli Society for Mathematical Statistics and Probability