Statistics Surveys

Log-concavity and strong log-concavity: A review

Adrien Saumard and Jon A. Wellner

Full-text: Open access

Abstract

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strong log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1965). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Article information

Source
Statist. Surv. Volume 8 (2014), 45-114.

Dates
First available in Project Euclid: 9 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1418134163

Digital Object Identifier
doi:10.1214/14-SS107

Mathematical Reviews number (MathSciNet)
MR3290441

Zentralblatt MATH identifier
06386273

Subjects
Primary: 60E15: Inequalities; stochastic orderings 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory

Keywords
Concave convex convolution inequalities log-concave monotone preservation strong log-concave

Citation

Saumard, Adrien; Wellner, Jon A. Log-concavity and strong log-concavity: A review. Statist. Surv. 8 (2014), 45--114. doi:10.1214/14-SS107. https://projecteuclid.org/euclid.ssu/1418134163.


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References

  • Adamczak, R., Litvak, A. E., Pajor, A. and Tomczak-Jaegermann, N. (2010). Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Amer. Math. Soc. 23 535–561.
  • Adamczak, R., Guédon, O., Latala, R., Litvak, A. E., Oleszkiewicz, K., Pajor, A. and Tomczak-Jaegermann, N. (2012). Moment estimates for convex measures. Electron. J. Probab. 17 no. 101, 19.
  • Alberti, G. and Ambrosio, L. (1999). A geometrical approach to monotone functions in $\mathbf{R}^{n}$. Math. Z. 230 259–316.
  • Alexandroff, A. D. (1939). Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 3–35.
  • An, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization. J. Economic Theory 80 350–369.
  • Andreu, F., Caselles, V. and Mazón, J. M. (2008). Some regularity results on the ‘relativistic’ heat equation. J. Differential Equations 245 3639–3663.
  • Artin, E. (1931). Einführung in die Theorie der Gammafunktion. Hamburger mathematische Einzelschriften, Heft II.
  • Artstein, S., Ball, K. M., Barthe, F. and Naor, A. (2004a). Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc. 17 975–982 (electronic).
  • Artstein, S., Ball, K. M., Barthe, F. and Naor, A. (2004b). On the rate of convergence in the entropic central limit theorem. Probab. Theory Related Fields 129 381–390.
  • Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econom. Theory 26 445–469.
  • Bakry, D. (1994). L’hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability Theory (Saint-Flour, 1992). Lecture Notes in Math. 1581 1–114. Springer, Berlin.
  • Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Berlin.
  • Balabdaoui, F. (2014). Global convergence of the log-concave MLE when the true distribution is geometric. J. Nonparametr. Stat. 26 21–59.
  • 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.
  • Balabdaoui, F. and Wellner, J. A. (2014). Chernoff’s density is log-concave. Bernoulli 20 231–244.
  • Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2013). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 769–790.
  • Ball, K. (2004). An elementary introduction to monotone transportation. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1850 41–52. Springer, Berlin.
  • Ball, K., Barthe, F. and Naor, A. (2003). Entropy jumps in the presence of a spectral gap. Duke Math. J. 119 41–63.
  • Barber, D. and Williams, C. K. I. (1997). Gaussian processes for Bayesian classification via hybrid Monte Carlo. In Advances in Neural Information Processing Systems (NIPS) 9. MIT Press.
  • Barbu, V. and Da Prato, G. (2008). The Kolmogorov operator associated with a stochastic variational inequality in $\mathbb{R}^{n}$ with convex potential. Rev. Roumaine Math. Pures Appl. 53 377–388.
  • Barbu, V. and Precupanu, T. (1986). Convexity and optimization in Banach spaces, second ed. Mathematics and Its Applications (East European Series) 10. D. Reidel Publishing Co., Dordrecht.
  • Barron, A. R. (1986). Entropy and the central limit theorem. Ann. Probab. 14 336–342.
  • Beran, R. and Millar, P. W. (1986). Confidence sets for a multivariate distribution. Ann. Statist. 14 431–443.
  • Bobkov, S. (1996). Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 35–48.
  • Bobkov, S. G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903–1921.
  • Bobkov, S. G. (2003). Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 37–43. Springer, Berlin.
  • Bobkov, S. G. (2008). A note on the distributions of the maximum of linear Bernoulli processes. Electron. Commun. Probab. 13 266–271.
  • Bobkov, S. G. (2010). Gaussian concentration for a class of spherically invariant measures. J. Math. Sci. (N. Y.) 167 326–339. Problems in mathematical analysis. No. 46.
  • Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
  • Bobkov, S. and Ledoux, M. (1997). Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 383–400.
  • Bobkov, S. G. and Ledoux, M. (2000). From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 1028–1052.
  • Bobkov, S. G. and Ledoux, M. (2009). Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 403–427.
  • Bobkov, S. and Ledoux, M. (2014). One-dimensional empirical measures, order statistics, and Kantorovich transport distances.
  • Bobkov, S. and Madiman, M. (2011). Concentration of the information in data with log-concave distributions. Ann. Probab. 39 1528–1543.
  • Bogachev, V. I. (1998). Gaussian measures. Mathematical Surveys and Monographs 62. American Mathematical Society, Providence, RI.
  • Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76. Springer-Verlag, New York.
  • Bondesson, L. (1997). On hyperbolically monotone densities. In Advances in the Theory and Practice of Statistics. Wiley Ser. Probab. Statist. Appl. Probab. Statist. 299–313. Wiley, New York.
  • Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
  • Borell, C. (1975). Convex set functions in $d$-space. Period. Math. Hungar. 6 111–136.
  • Borell, C. (1983). Convexity of measures in certain convex cones in vector space $\sigma$-algebras. Math. Scand. 53 125–144.
  • Boughorbel, S., Tarel, J. P. and Boujemaa, N. (2005). The LCCP for Optimizing Kernel Parameters for SVM. In Proceedings of International Conference on Artificial Neural Networks (ICANN’05) II 589–594. http://perso.lcpc.fr/tarel.jean-philippe/publis/icann05a.html.
  • Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press, Cambridge.
  • Brascamp, H. J. and Lieb, E. H. (1974). A logarithmic concavity theorem with some applications. Technical Report, Princeton University.
  • Brascamp, H. J. and Lieb, E. H. (1975). Some inequalities for Gaussian measures. In Functional Integration and Its Applications (A. M. Arthurs, ed.). Clarendon Press, Oxford.
  • Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Functional Analysis 22 366–389.
  • Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 375–417.
  • Brézis, H. (1973). Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).
  • Brooks, S. P. (1998). MCMC convergence diagnosis via multivariate bounds on log-concave densities. Ann. Statist. 26 398–433.
  • Brown, L. D. (1982). A proof of the central limit theorem motivated by the Cramér-Rao inequality. In Statistics and Probability: Essays in Honor of C. R. Rao. 141–148. North-Holland, Amsterdam.
  • Caffarelli, L. A. (1991). Some regularity properties of solutions of Monge Ampère equation. Comm. Pure Appl. Math. 44 965–969.
  • Caffarelli, L. A. (1992). The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5 99–104.
  • Caffarelli, L. A. (2000). Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys. 214 547–563.
  • Candès, E. J., Romberg, J. K. and Tao, T. (2006). Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 1207–1223.
  • Candès, E. J. and Tao, T. (2006). Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inform. Theory 52 5406–5425.
  • Carlen, E. A. and Cordero-Erausquin, D. (2009). Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities. Geom. Funct. Anal. 19 373–405.
  • Carlen, E. A., Cordero-Erausquin, D. and Lieb, E. H. (2013). Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures. Ann. Inst. Henri Poincaré Probab. Stat. 49 1–12.
  • Carlen, E. A., Lieb, E. H. and Loss, M. (2004). A sharp analog of Young’s inequality on $S^{N}$ and related entropy inequalities. J. Geom. Anal. 14 487–520.
  • Carlen, E. A. and Soffer, A. (1991). Entropy production by block variable summation and central limit theorems. Comm. Math. Phys. 140 339–371.
  • Cattiaux, P. and Guillin, A. (2013). Semi log-concave Markov diffusions. arXiv:0812.3141.
  • Chafaï, D., Guédon, O., Lecué, G. and Pajor, A. (2012). Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry. Panoramas et Synthèses [Panoramas and Syntheses] 37. Société Mathématique de France, Paris.
  • Chan, W., Park, D. H. and Proschan, F. (1989). Peakedness of weighted averages of jointly distributed random variables. In Contributions to Probability and Statistics. 58–62. Springer, New York.
  • Chapelle, O., Vapnik, V., Bousquet, O. and Mukherjee, S. (2002). Choosing multiple parameters for support vector machines. Machine Learning 46 131–159.
  • Cordero-Erausquin, D. (2002). Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161 257–269.
  • Cordero-Erausquin, D. (2005). On Berndtsson’s generalization of Prékopa’s theorem. Math. Z. 249 401–410.
  • Cordero-Erausquin, D. and Ledoux, M. (2010). The geometry of Euclidean convolution inequalities and entropy. Proc. Amer. Math. Soc. 138 2755–2769.
  • Cule, M., Gramacy, R. B. and Samworth, R. (2009). LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density. Journal of Statistical Software 29 1–20.
  • 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.
  • 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.
  • Das Gupta, S. (1980). Brunn-Minkowski inequality and its aftermath. J. Multivariate Anal. 10 296–318.
  • Das Gupta, S., Eaton, M. L., Olkin, I., Perlman, M., Savage, L. J. and Sobel, M. (1972). Inequalitites on the probability content of convex regions for elliptically contoured distributions. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory. 241–265. Univ. California Press, Berkeley, Calif.
  • Davidovič, J. S., Korenbljum, B. I. and Hacet, B. I. (1969). A certain property of logarithmically concave functions. Dokl. Akad. Nauk SSSR 185 1215–1218.
  • Devroye, L. (1984). A simple algorithm for generating random variates with a log-concave density. Computing 33 247–257.
  • Devroye, L. (2012). A note on generating random variables with log-concave densities. Statist. Probab. Lett. 82 1035–1039.
  • Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Probability and Mathematical Statistics. Academic Press Inc., Boston, MA.
  • Dinghas, A. (1957). Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus. Math. Z. 68 111–125.
  • Donoho, D. L. (2006). Compressed sensing. IEEE Trans. Inform. Theory 52 1289–1306.
  • Doss, C. H. and Wellner, J. A. (2013). Global rates of convergence of the MLEs of log-concave and s-concave densities. Technical Report No. 614, Department of Statistics, University of Washington. arXiv:1306.1438.
  • Dudley, R. M. (1977). On second derivatives of convex functions. Math. Scand. 41 159–174.
  • Dudley, R. M. (1980). Acknowledgment of priority: “On second derivatives of convex functions” [Math. Scand. 41 (1977), no. 1, 159–174; MR 58 #2250]. Math. Scand. 46 61.
  • 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.
  • Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
  • Efron, B. (1965). Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36 272–279.
  • Ehm, W. (1991). Binomial approximation to the Poisson binomial distribution. Statist. Probab. Lett. 11 7–16.
  • Ehrhard, A. (1983). Symétrisation dans l’espace de Gauss. Math. Scand. 53 281–301.
  • Fekete, M. (1912). Über ein problem von Laguerre. Rend. Circ., Mat. Palermo 34 89–100, 110–120.
  • Fort, G., Moulines, E., Roberts, G. O. and Rosenthal, J. S. (2003). On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40 123–146.
  • Fradelizi, M., Guédon, O. and Pajor, A. (2013). Spherical thin-shell concentration for convex measures. Technical Report, Université Paris-Est. arXiv:1306.6794v1.
  • Frieze, A., Kannan, R. and Polson, N. (1994a). Sampling from log-concave distributions. Ann. Appl. Probab. 4 812–837.
  • Frieze, A., Kannan, R. and Polson, N. (1994b). Correction: “Sampling from log-concave distributions”. Ann. Appl. Probab. 4 1255.
  • Frieze, A. and Kannan, R. (1999). Log-Sobolev inequalities and sampling from log-concave distributions. Ann. Appl. Probab. 9 14–26.
  • Gaenssler, P., Molnár, P. and Rost, D. (2007). On continuity and strict increase of the CDF for the sup-functional of a Gaussian process with applications to statistics. Results Math. 51 51–60.
  • Gardner, R. J. (2002). The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 355–405.
  • Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41 337–348.
  • Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields 16 635–736.
  • Guédon, O. (2012). Concentration phenomena in high dimensional geometry. In Proceedings of the Journées MAS 2012.
  • Guédon, O. and Milman, E. (2011). Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 1043–1068.
  • Gurvits, L. (2009). A short proof, based on mixed volumes, of Liggett’s theorem on the convolution of ultra-logconcave sequences. Electron. J. Combin. 16 Note 5, 5.
  • Hargé, G. (1999). A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 1939–1951.
  • Hargé, G. (2004). A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces. Probab. Theory Related Fields 130 415–440.
  • Hargé, G. (2008). Reinforcement of an inequality due to Brascamp and Lieb. J. Funct. Anal. 254 267–300.
  • Henningsson, T. and Astrom, K. J. (2006). Log-concave Observers. In Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems.
  • Ibragimov, I. A. (1956a). On the composition of unimodal distributions. Theor. Probability Appl. 1 255–266.
  • Ibragimov, I. A. (1956b). On the composition of unimodal distributions. Teor. Veroyatnost. i Primenen. 1 283–288.
  • Jensen, J. L. (1991). Uniform saddlepoint approximations and log-concave densities. J. Roy. Statist. Soc. Ser. B 53 157–172.
  • Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Statistical Science Series 16. The Clarendon Press Oxford University Press, New York. Oxford Science Publications.
  • Johnson, O. (2007). Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 791–802.
  • Johnson, O. and Barron, A. (2004). Fisher information inequalities and the central limit theorem. Probab. Theory Related Fields 129 391–409.
  • Johnson, O., Kontoyiannis, I. and Madiman, M. (2013). Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures. Discrete Appl. Math. 161 1232–1250.
  • Jylänki, P., Vanhatalo, J. and Vehtari, A. (2011). Robust Gaussian process regression with a Student-$t$ likelihood. J. Mach. Learn. Res. 12 3227–3257.
  • Kahn, J. and Neiman, M. (2011). A strong log-concavity property for measures on Boolean algebras. J. Combin. Theory Ser. A 118 1749–1760.
  • Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 541–559.
  • Kannan, R., Lovász, L. and Simonovits, M. (1997). Random walks and an $O^{*}(n^{5})$ volume algorithm for convex bodies. Random Structures Algorithms 11 1–50.
  • Karlin, S. (1968). Total Positivity. Vol. I. Stanford University Press, Stanford, Calif.
  • Keady, G. (1990). The persistence of logconcavity for positive solutions of the one-dimensional heat equation. J. Austral. Math. Soc. Ser. A 48 246–263.
  • Keilson, J. and Gerber, H. (1971). Some results for discrete unimodality. J. Amer. Statist. Assoc. 66 386–389.
  • Kelly, R. E. (1989). Stochastic reduction of loss in estimating normal means by isotonic regression. Ann. Statist. 17 937–940.
  • Kim, Y. H. and Milman, E. (2012). A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. Math. Ann. 354 827–862.
  • Kim, A. K. H. and Samworth, R. J. (2014). Global rates of convergence in log-concave density estimation. Technical Report, Statistical Laboratory, Cambridge University. Available as arXiv:1404.2298v1.
  • Klaassen, C. A. J. (1985). On an inequality of Chernoff. Ann. Probab. 13 966–974.
  • Klartag, B. and Milman, E. (2012). Inner regularization of log-concave measures and small-ball estimates. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 2050 267–278. Springer, Heidelberg.
  • Kolesnikov, A. V. (2001). On diffusion semigroups preserving the log-concavity. J. Funct. Anal. 186 196–205.
  • Koltchinskii, V. (2011). Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems. Lecture Notes in Mathematics 2033. Springer, Heidelberg. Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School].
  • Kontoyiannis, I., Harremoës, P. and Johnson, O. (2005). Entropy and the law of small numbers. IEEE Trans. Inform. Theory 51 466–472.
  • Korevaar, N. (1983a). Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32 73–81.
  • Korevaar, N. J. (1983b). Capillary surface continuity above irregular domains. Comm. Partial Differential Equations 8 213–245.
  • Ledoux, M. (1995). L’algèbre de Lie des gradients itérés d’un générateur markovien—développements de moyennes et entropies. Ann. Sci. École Norm. Sup. (4) 28 435–460.
  • Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 165–294. Springer, Berlin.
  • Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI.
  • Leindler, L. (1972). On a certain converse of Hölder’s inequality. II. Acta Sci. Math. (Szeged) 33 217–223.
  • Liggett, T. M. (1997). Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A 79 315–325.
  • Linnik, Y. V. (1959). An information theoretic proof of the central limit theorem with the Lindeberg condition. Theory Probab. Appl. 4 288–299.
  • Lovász, L. and Simonovits, M. (1993). Random walks in a convex body and an improved volume algorithm. Random Structures Algorithms 4 359–412.
  • Lovasz, L. and Vempala, S. (2006). Fast algorithms for logconcave functions: Sampling, rounding, integration and optimization. In Foundations of Computer Science, 2006. FOCS ’06. 47th Annual IEEE Symposium on. 57– 68.
  • Lovász, L. and Vempala, S. (2007). The geometry of logconcave functions and sampling algorithms. Random Structures Algorithms 30 307–358.
  • Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering 143. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York.
  • Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, second ed. Springer Series in Statistics. Springer, New York.
  • Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188–197.
  • Maurey, B. (2005). Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles. Astérisque 299 Exp. No. 928, vii, 95–113. Séminaire Bourbaki. Vol. 2003/2004.
  • McCann, R. J. (1995). Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 309–323.
  • Menz, G. and Otto, F. (2013). Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Annals of Probability 41 2182–2224.
  • Minka, T. P. (2001). A family of algorithms for approximate Bayesian inference. PhD thesis, Massachusetts Institute of Technology.
  • Nayar, P. and Oleszkiewicz, K. (2012). Khinchine type inequalities with optimal constants via ultra log-concavity. Positivity 16 359–371.
  • Neal, R. M. (2003). Slice sampling. Ann. Statist. 31 705–767. With discussions and a rejoinder by the author.
  • Niculescu, C. P. and Persson, L. E. (2006). Convex Functions and Their Applications. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23. Springer, New York. A contemporary approach.
  • Olkin, I. and Tong, Y. L. (1988). Peakedness in multivariate distributions. In Statistical Decision Theory and Related Topics, IV, Vol. 2 (West Lafayette, Ind., 1986). 373–383. Springer, New York.
  • Paninski, L. (2004). Log-concavity results on Gaussian process methods for supervised and unsupervised learning. In Advances in Neural Information Processing Systems. 1025–1032.
  • Pitt, L. D. (1977). A Gaussian correlation inequality for symmetric convex sets. Ann. Probability 5 470–474.
  • Polson, N. G. (1996). Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics, 5 (Alicante, 1994). Oxford Sci. Publ. 297–321. Oxford Univ. Press, New York.
  • Port, S. C. and Stone, C. J. (1974). Fisher information and the Pitman estimator of a location parameter. Ann. Statist. 2 225–247.
  • Prékopa, A. (1971). Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 301–316.
  • Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 335–343.
  • Prékopa, A. (1995). Stochastic Programming. Mathematics and Its Applications 324. Kluwer Academic Publishers Group, Dordrecht.
  • Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist. 36 1703–1706.
  • Rinott, Y. (1976). On convexity of measures. Ann. Probability 4 1020–1026.
  • Roberts, G. O. and Rosenthal, J. S. (2002). The polar slice sampler. Stoch. Models 18 257–280.
  • Rockafellar, R. T. and Wets, R. J. B. (1998). Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin.
  • Rudin, W. (1987). Real and Complex Analysis, Third ed. McGraw-Hill Book Co., New York.
  • Rudolf, D. (2012). Explicit error bounds for Markov chain Monte Carlo. Dissertationes Math. (Rozprawy Mat.) 485 1–93.
  • Saumard, A. and Wellner, J. A. (2015). Efron’s monotonicity theorem and asymmetric Brascamp-Lieb type inequalities. Technical Report, Department of Statistics, University of Washington. In preparation.
  • Schoenberg, I. J. (1951). On Pólya frequency functions. I. The totally positive functions and their Laplace transforms. J. Analyse Math. 1 331–374.
  • Seeger, M. (2004). Gaussian processes for machine learning. International Journal of Neural Systems 14 69–106.
  • Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 3751–3781. With supplementary material available online.
  • Sherman, S. (1955). A theorem on convex sets with applications. Ann. Math. Statist. 26 763–767.
  • Simon, B. (2011). Convexity. Cambridge Tracts in Mathematics 187. Cambridge University Press, Cambridge. An analytic viewpoint.
  • Tsirel’son, V. S. (1975). The density of the distribution of the maximum of a Gaussian process. Theory Probab. Appl. 20 847–856.
  • 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.
  • van der Vaart, A. and van Zanten, H. (2011). Information rates of nonparametric Gaussian process methods. J. Mach. Learn. Res. 12 2095–2119.
  • Vapnik, V. N. (2000). The Nature of Statistical Learning Theory, second ed. Statistics for Engineering and Information Science. Springer-Verlag, New York.
  • Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI.
  • Villani, C. (2009). Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin. Old and new.
  • Walkup, D. W. (1976). Pólya sequences, binomial convolution and the union of random sets. J. Appl. Probability 13 76–85.
  • Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
  • Walther, G. (2009). Inference and modeling with log-concave distributions. Statist. Sci. 24 319–327.
  • Wang, Z. and Louis, T. A. (2003). Matching conditional and marginal shapes in binary random intercept models using a bridge distribution function. Biometrika 90 765–776.
  • Wellner, J. A. (2013). Strong log-concavity is preserved by convolution. In High Dimensional Probability VI: The Banff Volume. Progresss in Probability 66 95–102. Birkhauser, Basel.
  • Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. J. Appl. Probab. 22 619–633.
  • Yuille, A. and Rangarajan, A. (2003). The concave-convex procedure (CCCP). Neural Computation 15 915–936.
  • Zhang, Z., Dai, G. and Jordan, M. I. (2011). Bayesian generalized kernel mixed models. J. Mach. Learn. Res. 12 111–139.