A Bayesian nonparametric approach to log-concave density estimation. (English) Zbl 1466.62285

Summary: The estimation of a log-concave density on \(\mathbb{R}\) is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.


62G07 Density estimation
62F15 Bayesian inference
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI arXiv Euclid


[1] Balabdaoui, F. and Doss, C.R. (2018). Inference for a two-component mixture of symmetric distributions under log-concavity. Bernoulli 24 1053-1071. · Zbl 1419.62059
[2] Balabdaoui, F., Rufibach, K. and Wellner, J.A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299-1331. · Zbl 1160.62008
[3] Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970-981. · Zbl 0888.62033
[4] Cule, M. and Samworth, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Stat. 4 254-270. · Zbl 1329.62183
[5] Cule, M., Samworth, R. and Stewart, M. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 545-607. · Zbl 1411.62055
[6] Doss, C.R. and Wellner, J.A. (2016). Global rates of convergence of the MLEs of log-concave and \(s\)-concave densities. Ann. Statist. 44 954-981. · Zbl 1338.62101
[7] Doss, C.R. and Wellner, J.A. (2019). Inference for the mode of a log-concave density. Ann. Statist. 47 2950-2976. · Zbl 1439.62098
[8] Doss, C.R. and Wellner, J.A. (2019). Univariate log-concave density estimation with symmetry or modal constraints. Electron. J. Stat. 13 2391-2461. · Zbl 1422.62137
[9] Dümbgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli 15 40-68. · Zbl 1200.62030
[10] Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702-730. · Zbl 1216.62023
[11] Ghosal, S., Ghosh, J.K. and van der Vaart, A.W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. · Zbl 1105.62315
[12] Ghosal, S. and van der Vaart, A. (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Statist. 35 697-723. · Zbl 1117.62046
[13] Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge Series in Statistical and Probabilistic Mathematics 44. Cambridge: Cambridge Univ. Press. · Zbl 1376.62004
[14] Giné, E. and Nickl, R. (2011). Rates of contraction for posterior distributions in \(L^r\)-metrics, \(1\leq r\leq\infty \). Ann. Statist. 39 2883-2911. · Zbl 1246.62095
[15] Groeneboom, P. and Jongbloed, G. (2014). Nonparametric Estimation Under Shape Constraints: Estimators, Algorithms and Asymptotics. Cambridge Series in Statistical and Probabilistic Mathematics 38. New York: Cambridge Univ. Press. · Zbl 1338.62008
[16] Han, Q. (2017). Bayes model selection. ArXiv E-prints.
[17] Hannah, L.A. and Dunson, D.B. (2011). Bayesian nonparametric multivariate convex regression. ArXiv e-prints.
[18] Has’minskiĭ, R.Z. (1979). Lower bound for the risks of nonparametric estimates of the mode. In Contributions to Statistics 91-97. Dordrecht: Reidel.
[19] Ibragimov, I.A. (1956). On the composition of unimodal distributions. Teor. Veroyatn. Primen. 1 283-288. · Zbl 0073.12501
[20] Khazaei, S. and Rousseau, J. (2010). Bayesian nonparametric inference of decreasing densities. In 42èmes Journées de Statistique. Marseille, France.
[21] Kim, A.K.H., Guntuboyina, A. and Samworth, R.J. (2018). Adaptation in log-concave density estimation. Ann. Statist. 46 2279-2306. · Zbl 1408.62062
[22] Kim, A.K.H. and Samworth, R.J. (2016). Global rates of convergence in log-concave density estimation. Ann. Statist. 44 2756-2779. · Zbl 1360.62157
[23] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. New York: Springer. · Zbl 0605.62002
[24] Mariucci, E., Ray, K. and Szabó, B. (2020). Supplement to “A Bayesian nonparametric approach to log-concave density estimation.” https://doi.org/10.3150/19-BEJ1139SUPP.
[25] Müller, S. and Rufibach, K. (2009). Smooth tail-index estimation. J. Stat. Comput. Simul. 79 1155-1167. · Zbl 1179.62075
[26] Ray, K. (2013). Bayesian inverse problems with non-conjugate priors. Electron. J. Stat. 7 2516-2549. · Zbl 1294.62107
[27] Reiss, M. and Schmidt-Hieber, J. (2019). Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery. Ann. Statist. To appear. Available at arXiv:1809.04140.
[28] Salomond, J.-B. (2014). Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints. Electron. J. Stat. 8 1380-1404. · Zbl 1298.62064
[29] Samworth, R.J. and Yuan, M. (2012). Independent component analysis via nonparametric maximum likelihood estimation. Ann. Statist. 40 2973-3002. · Zbl 1296.62084
[30] Seregin, A. and Wellner, J.A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 3751-3781. With supplementary material available online. · Zbl 1204.62058
[31] Shively, T.S., Sager, T.W. and Walker, S.G. (2009). A Bayesian approach to non-parametric monotone function estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 159-175. · Zbl 1231.62058
[32] Shively, T.S., Walker, S.G. and Damien, P. (2011). Nonparametric function estimation subject to monotonicity, convexity and other shape constraints. J. Econometrics 161 166-181. · Zbl 1441.62870
[33] Szabó, B., van der Vaart, A.W. and van Zanten, J.H. (2015). Frequentist coverage of adaptive nonparametric Bayesian credible sets. Ann. Statist. 43 1391-1428. · Zbl 1317.62040
[34] van de Geer, S.A. (2000). Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge: Cambridge Univ. Press.
[35] van der Vaart, A.W. and van Zanten, J.H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 1435-1463. · Zbl 1141.60018
[36] Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508-513. · Zbl 1073.62533
[37] Walther, G. (2009). Inference and modeling with log-concave distributions. Statist. Sci. 24 319-327. · Zbl 1329.62192
[38] Williamson, R.E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189-207. · Zbl 0070.28501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.