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2018 Normalizing constants of log-concave densities
Nicolas Brosse, Alain Durmus, Éric Moulines
Electron. J. Statist. 12(1): 851-889 (2018). DOI: 10.1214/18-EJS1411


We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $\pi =\mathrm{e}^{-U}/Z$ w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm [15]. Polynomial bounds in the dimension $d$ are obtained with an exponent that depends on the assumptions made on $U$. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.


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Nicolas Brosse. Alain Durmus. Éric Moulines. "Normalizing constants of log-concave densities." Electron. J. Statist. 12 (1) 851 - 889, 2018.


Received: 1 July 2017; Published: 2018
First available in Project Euclid: 5 March 2018

zbMATH: 06864479
MathSciNet: MR3770890
Digital Object Identifier: 10.1214/18-EJS1411

Primary: 60F25 , 62L10 , 65C05
Secondary: 60J05 , 65C40 , 74G10 , 74G15

Keywords: annealed importance sampling , Bayes factor , Normalizing constants , Unadjusted Langevin Algorithm


Vol.12 • No. 1 • 2018
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