# Normalizing constants of log-concave densities

1 XPOP - Modélisation en pharmacologie de population
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : 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 (Durmus, A. and Moulines, E. (2016). High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm). 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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Cited literature [42 references]

https://hal.inria.fr/hal-01648666
Contributor : Nicolas Brosse <>
Submitted on : Monday, November 27, 2017 - 11:25:23 PM
Last modification on : Friday, July 31, 2020 - 10:44:09 AM

### Citation

Nicolas Brosse, Alain Durmus, Éric Moulines. Normalizing constants of log-concave densities. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.851-889. ⟨10.1214/18-EJS1411⟩. ⟨hal-01648666v1⟩

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