Electronic Journal of Statistics

Normalizing constants of log-concave densities

Nicolas Brosse, Alain Durmus, and Éric Moulines

Full-text: Open access

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 [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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1520240451

Digital Object Identifier
doi:10.1214/18-EJS1411

Mathematical Reviews number (MathSciNet)
MR3770890

Zentralblatt MATH identifier
06864479

Subjects
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

Keywords
Normalizing constants Bayes factor annealed importance sampling Unadjusted Langevin Algorithm

Rights
Creative Commons Attribution 4.0 International License.

Citation

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


Export citation

References

  • [1] C. Andrieu, J. Ridgway, and N. Whiteley. Sampling normalizing constants in high dimensions using inhomogeneous diffusions., ArXiv e-prints, Dec. 2016.
  • [2] D. Ardia, N. Baştürk, L. Hoogerheide, and H. K. Van Dijk. A comparative study of Monte Carlo methods for efficient evaluation of marginal likelihood., Computational Statistics & Data Analysis, 56(11) :3398–3414, 2012.
  • [3] R. Balian., From microphysics to macrophysics: methods and applications of statistical physics, volume 1. Springer Science & Business Media, 2007.
  • [4] G. Behrens, N. Friel, and M. Hurn. Tuning tempered transitions., Statistics and Computing, 22(1):65–78, 2012. URL http://dx.doi.org/10.1007/s11222-010-9206-z.
  • [5] A. Beskos, D. O. Crisan, A. Jasra, and N. Whiteley. Error bounds and normalising constants for sequential Monte Carlo samplers in high dimensions., Adv. in Appl. Probab., 46(1):279–306, 03 2014. URL http://dx.doi.org/10.1239/aap/1396360114.
  • [6] S. Boucheron, G. Lugosi, and P. Massart., Concentration inequalities: a nonasymptotic theory of independence. Oxford university press, 2013.
  • [7] S. Brazitikos, A. Giannopoulos, P. Valettas, and B.-H. Vritsiou., Geometry of isotropic convex bodies, volume 196. American Mathematical Society Providence, 2014.
  • [8] N. Brosse, A. Durmus, and E. Moulines. Supplement to “Normalizing constants of log-concave densities”. DOI:, 10.1214/18-EJS1411SUPP, 2018.
  • [9] M. Chen, Q. Shao, and J. Ibrahim., Monte Carlo methods in Bayesian computation. Springer, New York, 2000.
  • [10] B. Cousins and S. Vempala. Bypassing KLS: Gaussian cooling and an $\textO^*(n^3)$ volume algorithm. In, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 539–548. ACM, 2015.
  • [11] A. S. Dalalyan. Theoretical guarantees for approximate sampling from smooth and log-concave densities., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2016.
  • [12] P. Del Moral., Feynman-Kac formulae. Probability and its Applications (New York). Springer-Verlag, New York, 2004. ISBN 0-387-20268-4. URL https://doi.org/10.1007/978-1-4684-9393-1. Genealogical and interacting particle systems with applications.
  • [13] P. Del Moral, A. Jasra, K. Law, and Y. Zhou. Multilevel sequential Monte Carlo samplers for normalizing constants., ArXiv e-prints, Mar. 2016.
  • [14] A. Durmus and E. Moulines. Non-asymptotic convergence analysis for the unadjusted Langevin algorithm. July, 2015.
  • [15] A. Durmus and E. Moulines. High-dimensional Bayesian inference via the unadjusted Langevin algorithm. May, 2016.
  • [16] R. Dutta, J. K. Ghosh, et al. Bayes model selection with path sampling: factor models and other examples., Statistical Science, 28(1):95–115, 2013.
  • [17] M. Dyer and A. Frieze. Computing the volume of convex bodies: a case where randomness provably helps., Probabilistic combinatorics and its applications, 44:123–170, 1991.
  • [18] D. L. Ermak. A computer simulation of charged particles in solution. i. technique and equilibrium properties., The Journal of Chemical Physics, 62(10) :4189–4196, 1975.
  • [19] L. C. Evans and R. F. Gariepy., Measure theory and fine properties of functions. CRC press, 2015.
  • [20] N. Friel and J. Wyse. Estimating the evidence–a review., Statistica Neerlandica, 66(3):288–308, 2012.
  • [21] N. Friel, M. Hurn, and J. Wyse. Improving power posterior estimation of statistical evidence., Statistics and Computing, 24(5):709–723, 2014. ISSN 1573-1375. URL http://dx.doi.org/10.1007/s11222-013-9397-1.
  • [22] A. Gelman and X.-L. Meng. Simulating normalizing constants: From importance sampling to bridge sampling to path sampling., Statistical science, pages 163–185, 1998.
  • [23] M. Huber. Approximation algorithms for the normalizing constant of Gibbs distributions., Ann. Appl. Probab., 25(2):974–985, 04 2015. URL http://dx.doi.org/10.1214/14-AAP1015.
  • [24] C. Jarzynski. Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach., Physical Review E, 56(5) :5018, 1997.
  • [25] A. Jasra, C. C. Holmes, and D. A. Stephens. Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling., Statist. Sci., 20(1):50–67, 02 2005. URL https://doi.org/10.1214/088342305000000016.
  • [26] A. Jasra, K. Kamatani, P. P. Osei, and Y. Zhou. Multilevel particle filters: normalizing constant estimation., Statistics and Computing, pages 1–14, 2016. ISSN 1573-1375. URL http://dx.doi.org/10.1007/s11222-016-9715-5.
  • [27] M. R. Jerrum, L. G. Valiant, and V. V. Vazirani. Random generation of combinatorial structures from a uniform distribution., Theoretical Computer Science, 43:169–188, 1986. ISSN 0304-3975. URL http://www.sciencedirect.com/science/article/pii/030439758690174X.
  • [28] A. Joulin and Y. Ollivier. Curvature, concentration and error estimates for Markov chain Monte Carlo., Ann. Probab., 38(6) :2418–2442, 11 2010. URL http://dx.doi.org/10.1214/10-AOP541.
  • [29] K. H. Knuth, M. Habeck, N. K. Malakar, A. M. Mubeen, and B. Placek. Bayesian evidence and model selection., Digital Signal Processing, 47:50–67, 2015. ISSN 1051-2004. URL http://www.sciencedirect.com/science/article/pii/S1051200415001980. Special Issue in Honour of William J. (Bill) Fitzgerald.
  • [30] T. Lelièvre, G. Stoltz, and M. Rousset., Free energy computations: A mathematical perspective. World Scientific, 2010.
  • [31] J.-M. Marin and C. P. Robert. Importance sampling methods for Bayesian discrimination between embedded models., arXiv preprint arXiv :0910.2325, 2009.
  • [32] S. P. Meyn and R. L. Tweedie. Stability of Markovian processes iii: Foster-Lyapunov criteria for continuous-time processes., Advances in Applied Probability, pages 518–548, 1993.
  • [33] P. D. Moral, A. Doucet, and A. Jasra. Sequential Monte Carlo samplers., Journal of the Royal Statistical Society. Series B (Statistical Methodology), 68(3):411–436, 2006. ISSN 13697412, 14679868. URL http://www.jstor.org/stable/3879283.
  • [34] R. M. Neal. Annealed importance sampling., Statistics and Computing, 11(2):125–139, 2001. ISSN 1573-1375. URL http://dx.doi.org/10.1023/A:1008923215028.
  • [35] Y. Nesterov., Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.
  • [36] W. Niemiro and P. Pokarowski. Fixed precision MCMC estimation by median of products of averages., J. Appl. Probab., 46(2):309–329, 06 2009. URL http://dx.doi.org/10.1239/jap/1245676089.
  • [37] C. J. Oates, T. Papamarkou, and M. Girolami. The controlled thermodynamic integral for Bayesian model evidence evaluation., Journal of the American Statistical Association, 111(514):634–645, 2016. URL http://dx.doi.org/10.1080/01621459.2015.1021006.
  • [38] G. Parisi. Correlation functions and computer simulations., Nuclear Physics B, 180:378–384, 1981.
  • [39] M. Pereyra. Maximum-a-posteriori estimation with Bayesian confidence regions., arXiv preprint arXiv :1602.08590, 2016.
  • [40] F. Proschan and J. Sethuraman. Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability., Journal of Multivariate Analysis, 6(4):608–616, 1976. ISSN 0047-259X. URL http://www.sciencedirect.com/science/article/pii/0047259X76900087.
  • [41] R Core Team., R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2018. URL https://www.R-project.org/.
  • [42] S. Richardson and P. J. Green. On Bayesian analysis of mixtures with an unknown number of components (with discussion)., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59(4):731–792, 1997. ISSN 1467-9868. URL http://dx.doi.org/10.1111/1467-9868.00095.
  • [43] C. Robert., The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science & Business Media, 2007.
  • [44] G. O. Roberts and R. L. Tweedie. Exponential convergence of Langevin distributions and their discrete approximations., Bernoulli, 2(4):341–363, 1996. ISSN 1350-7265. URL http://dx.doi.org/10.2307/3318418.
  • [45] J. P. Valleau and D. N. Card. Monte Carlo estimation of the free energy by multistage sampling., The Journal of Chemical Physics, 57(12) :5457–5462, 1972. URL http://dx.doi.org/10.1063/1.1678245.
  • [46] C. Villani., Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.
  • [47] E. J. Williams and E. Williams., Regression analysis, volume 14. wiley New York, 1959.
  • [48] J. Wyse. Estimating the statistical evidence - a review, 2011. URL, https://sites.google.com/site/jsnwyse/code.
  • [49] Y. Zhou, A. M. Johansen, and J. A. Aston. Towards automatic model comparison: an adaptive sequential Monte Carlo approach., Journal of Computational and Graphical Statistics, (just-accepted), 2015.

Supplemental materials