Electronic Journal of Statistics

Normalizing constants of log-concave densities

Nicolas Brosse, Alain Durmus, and Éric Moulines

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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.

Article information

Electron. J. Statist., Volume 12, Number 1 (2018), 851-889.

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

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 60F25: $L^p$-limit theorems 62L10: Sequential analysis
Secondary: 65C40: Computational Markov chains 60J05: Discrete-time Markov processes on general state spaces 74G10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) 74G15: Numerical approximation of solutions

Normalizing constants Bayes factor annealed importance sampling Unadjusted Langevin Algorithm

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Brosse, Nicolas; Durmus, Alain; Moulines, Éric. Normalizing constants of log-concave densities. Electron. J. Statist. 12 (2018), no. 1, 851--889. doi:10.1214/18-EJS1411. https://projecteuclid.org/euclid.ejs/1520240451

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