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Toward Machine Wald

  • Houman OwhadiEmail author
  • Clint Scovel
Reference work entry

Abstract

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed by humans because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to think as humans, especially when faced with uncertainty, is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well-posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tends to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with decision theory, machine learning, Bayesian inference, stochastic optimization, robust optimization, optimal uncertainty quantification, and information-based complexity.

Keywords

Abraham Wald Decision theory Machine learning Uncertainty quantification Game theory 

References

  1. 1.
    Richardson, L.F.: Weather Prediction by Numerical Process. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1922)zbMATHGoogle Scholar
  2. 2.
    Ackerman, N.L., Freer, C.E., Roy, D.M.: On the computability of conditional probability. arXiv:1005.3014 (2010)Google Scholar
  3. 3.
    Adams, M., Lashgari, A., Li, B., McKerns, M., Mihaly, J.M., Ortiz, M., Owhadi, H., Rosakis, A.J., Stalzer, M., Sullivan, T.J.: Rigorous model-based uncertainty quantification with application to terminal ballistics. Part II: systems with uncontrollable inputs and large scatter. J. Mech. Phys. Solids 60(5), 1002–1019 (2012)Google Scholar
  4. 4.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc. 6(2), 170–176 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Belot, G.: Bayesian orgulity. Philos. Sci. 80(4), 483–503 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)zbMATHCrossRefGoogle Scholar
  8. 8.
    Ben-Tal, A., Hochman, E.: More bounds on the expectation of a convex function of a random variable. J. Appl. Probab. 9, 803–812 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bentkus, V.: A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand. Liet. Mat. Rink. 42(3), 332–342 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bentkus, V.: On Hoeffding’s inequalities. Ann. Probab. 32(2), 1650–1673 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bentkus, V., Geuze, G.D.C., van Zuijlen, M.C.A.: Optimal Hoeffding-like inequalities under a symmetry assumption. Statistics 40(2), 159–164 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bernstein, S.N.: Collected Works. Izdat. “Nauka”, Moscow (1964)Google Scholar
  14. 14.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (electronic) (2005)Google Scholar
  16. 16.
    Birge, J.R., Wets, R.J.-B.: Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse. Math. Prog. Stud. 27, 54–102 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Blackwell, D.: Equivalent comparisons of experiments. Ann. Math. Stat. 24(2), 265–272 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Bogachev, V.I.: Measure Theory, vol. II. Springer, Berlin (2007)zbMATHCrossRefGoogle Scholar
  19. 19.
    Boţ, R.I., Lorenz, N., Wanka, G.: Duality for linear chance-constrained optimization problems. J. Korean Math. Soc. 47(1), 17–28 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Boucheron, S., Lugosi, G., Massart, P.: A sharp concentration inequality with applications. Random Struct. Algorithms 16(3), 277–292 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Brown, L.D.: Minimaxity, more or less. In: Gupta, S.S., Berger, J.O. (eds.) Statistical Decision Theory and Related Topics V, pp. 1–18. Springer, New York (1994)Google Scholar
  22. 22.
    Brown, L.D.: An essay on statistical decision theory. J. Am. Stat. Assoc. 95(452), 1277–1281 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Castillo, I., Nickl, R.: Nonparametric Bernstein–von Mises theorems in Gaussian white noise. Ann. Stat. 41(4), 1999–2028 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Chen, W., Sim, M., Sun, J., Teo, C.-P.: From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper. Res. 58(2), 470–485 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes.: General Theory and Structure. Probability and its Applications (New York), vol. II, 2nd edn. Springer, New York (2008)Google Scholar
  26. 26.
    Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1, 197–206 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Diaconis, P., Freedman, D.A.: On the consistency of Bayes estimates. Ann. Stat. 14(1), 1–67 (1986). With a discussion and a rejoinder by the authorsGoogle Scholar
  28. 28.
    Doob, J.L.: Application of the theory of martingales. In: Le Calcul des Probabilités et ses Applications, Colloques Internationaux du Centre National de la Recherche Scientifique, vol. 13, pp. 23–27. Centre National de la Recherche Scientifique, Paris (1949)Google Scholar
  29. 29.
    Doob, J.L.: Measure Theory. Graduate Texts in Mathematics, vol. 143. Springer, New York (1994)Google Scholar
  30. 30.
    Drenick, R.F.: Aseismic design by way of critical excitation. J. Eng. Mech. Div. Am. Soc. Civ. Eng. 99(4), 649–667 (1973)Google Scholar
  31. 31.
    Dubins, L.E.: On extreme points of convex sets. J. Math. Anal. Appl. 5(2), 237–244 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002). Revised reprint of the 1989 originalGoogle Scholar
  33. 33.
    Dvoretzky, A., Wald, A., Wolfowitz, J.: Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. Ann. Math. Stat. 22(1), 1–21 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Edmundson, H.P.: Bounds on the expectation of a convex function of a random variable. Technical report, DTIC Document (1957)Google Scholar
  35. 35.
    Elishakoff, I., Ohsaki, M.: Optimization and Anti-optimization of Structures Under Uncertainty. World Scientific, London (2010)zbMATHCrossRefGoogle Scholar
  36. 36.
    Ermoliev, Y., Gaivoronski, A., Nedeva, C.: Stochastic optimization problems with incomplete information on distribution functions. SIAM J. Control Optim. 23(5), 697–716 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Fisher, R.: The Design of Experiments. Oliver and Boyd, Edinburgh (1935)Google Scholar
  38. 38.
    Fisher, R.: Statistical methods and scientific induction. J. R. Stat. Soc. Ser. B. 17, 69–78 (1955)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A 222, 309–368 (1922)zbMATHCrossRefGoogle Scholar
  40. 40.
    Fisher, R.A.: “Student”. Ann. Eugen. 9(1), 1–9 (1939)zbMATHCrossRefGoogle Scholar
  41. 41.
    Frauendorfer, K.: Solving SLP recourse problems with arbitrary multivariate distributions-the dependent case. Math. Oper. Res. 13(3), 377–394 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Freedman, D.A.: On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Stat. 34, 1386–1403 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Freedman, D.A.: On the Bernstein-von Mises theorem with infinite-dimensional parameters. Ann. Stat. 27(4), 1119–1140 (1999)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Gaivoronski, A.A.: A numerical method for solving stochastic programming problems with moment constraints on a distribution function. Ann. Oper. Res. 31(1), 347–369 (1991)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Gassmann, H., Ziemba, W.T.: A tight upper bound for the expectation of a convex function of a multivariate random variable. In: Stochastic Programming 84 Part I. Mathematical Programming Study, vol. 27, pp. 39–53. Springer, Berlin (1986)Google Scholar
  46. 46.
    Geoffrion, A.M.: Generalized Benders decomposition. JOTA 10(4), 237–260 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18(2), 141–153 (1989)zbMATHCrossRefGoogle Scholar
  48. 48.
    Godwin, H.J.: On generalizations of Tchebychef’s inequality. J. Am. Stat. Assoc. 50(271), 923–945 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4, part 1), 902–917 (2010)Google Scholar
  50. 50.
    Halmos, P.R., Savage, L.J.: Application of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Stat. 20(2), 225–241 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Han, S., Tao, M., Topcu, U., Owhadi, H., Murray, R.M.: Convex optimal uncertainty quantification. SIAM J. Optim. 25(23), 1368–1387 (2015). arXiv:1311.7130Google Scholar
  52. 52.
    Han, S., Topcu, U., Tao, M., Owhadi, H., Murray, R.: Convex optimal uncertainty quantification: algorithms and a case study in energy storage placement for power grids. In: American Control Conference (ACC), 2013, Washington, DC, pp. 1130–1137. IEEE (2013)Google Scholar
  53. 53.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Program. 151(1), 35–62 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Hoeffding, W.: On the distribution of the number of successes in independent trials. Ann. Math. Stat. 27(3), 713–721 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Hotelling, H.: Abraham Wald. Am. Stat. 5(1), 18–19 (1951)CrossRefGoogle Scholar
  56. 56.
    Huang, C.C., Vertinsky, I., Ziemba, W.T.: Sharp bounds on the value of perfect information. Oper. Res. 25(1), 128–139 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Huang, C.C., Ziemba, W.T., Ben-Tal, A.: Bounds on the expectation of a convex function of a random variable: with applications to stochastic programming. Oper. Res. 25(2), 315–325 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Huber, P.J.: The 1972 Wald lecture- Robust statistics: a review. Ann. Math. Stat. 1041–1067 (1972)Google Scholar
  60. 60.
    Isii, K.: On a method for generalizations of Tchebycheff’s inequality. Ann. Inst. Stat. Math. Tokyo 10(2), 65–88 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Isii, K.: The extrema of probability determined by generalized moments. I. Bounded random variables. Ann. Inst. Stat. Math. 12(2), 119–134; errata, 280 (1960)Google Scholar
  62. 62.
    Isii, K.: On sharpness of Tchebycheff-type inequalities. Ann. Inst. Stat. Math. 14(1):185–197, 1962/1963.Google Scholar
  63. 63.
    Jaynes, E.T.: Probability Theory. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  64. 64.
    Joe, H.: Majorization, randomness and dependence for multivariate distributions. Ann. Probab. 15(3), 1217–1225 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Johnstone, I.M.: High dimensional Bernstein–von Mises: simple examples. In Borrowing Strength: Theory Powering Applications—A Festschrift for Lawrence D. Brown, volume 6 of Inst. Math. Stat. Collect., pages 87–98. Inst. Math. Statist., Beachwood, OH (2010)Google Scholar
  66. 66.
    Kac, M., Slepian, D.: Large excursions of Gaussian processes. Ann. Math. Stat. 30, 1215–1228 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Kall, P.: Stochastric programming with recourse: upper bounds and moment problems: a review. Math. Res. 45, 86–103 (1988)Google Scholar
  68. 68.
    Kallenberg, O.: Random Measures. Akademie-Verlag, Berlin (1975) Schriftenreihe des Zentralinstituts für Mathematik und Mechanik bei der Akademie der Wissenschaften der DDR, Heft 23.Google Scholar
  69. 69.
    Kamga, P.-H.T., Li, B., McKerns, M., Nguyen, L.H., Ortiz, M., Owhadi, H., Sullivan, T.J.: Optimal uncertainty quantification with model uncertainty and legacy data. J. Mech. Phys. Solids 72, 1–19 (2014)CrossRefGoogle Scholar
  70. 70.
    Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics, vol. XV. Interscience Publishers/Wiley, New York/London/Sydney (1966)Google Scholar
  71. 71.
    Kendall, D.G.: Simplexes and vector lattices. J. Lond. Math. Soc. 37(1), 365–371 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Kidane, A.A., Lashgari, A., Li, B., McKerns, M., Ortiz, M., Owhadi, H., Ravichandran, G., Stalzer, M., Sullivan, T.J.: Rigorous model-based uncertainty quantification with application to terminal ballistics. Part I: Systems with controllable inputs and small scatter. J. Mech. Phys. Solids 60(5), 983–1001 (2012)Google Scholar
  73. 73.
    Kiefer, J.: Optimum experimental designs. J. R. Stat. Soc. Ser. B 21, 272–319 (1959)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Kiefer, J.: Collected Works, vol. III. Springer, New York (1985)Google Scholar
  75. 75.
    Kleijn, B.J.K., van der Vaart, A.W.: The Bernstein-Von-Mises theorem under misspecification. Electron. J. Stat. 6, 354–381 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea Publishing Co., New York (1956). Translation edited by Nathan Morrison, with an added bibliography by A. T. Bharucha-ReidGoogle Scholar
  77. 77.
    Kreĭn, M.G.: The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development. In: Dynkin, E.B. (ed.) Eleven Papers on Analysis, Probability, and Topology, American Mathematical Society Translations, Series 2, vol. 12, pp. 1–122. American Mathematical Society, New York (1959)Google Scholar
  78. 78.
    Kurz, H.D., Salvadori, N.: Understanding ‘Classical’ Economics: Studies in Long Period Theory. Routledge, London/New York (2002)Google Scholar
  79. 79.
    Laird, N.M.: A conversation with F. N. David. Stat. Sci. 4, 235–246 (1989)CrossRefGoogle Scholar
  80. 80.
    Le Cam, L.: On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates. Univ. Calif. Publ. Stat. 1, 277–329 (1953)MathSciNetGoogle Scholar
  81. 81.
    Le Cam, L.: An extension of Wald’s theory of statistical decision functions. Ann. Math. Stat. 26, 69–81 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Le Cam, L.: Sufficiency and approximate sufficiency. Ann. Math. Stat. 35, 1419–1455 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. Springer, New York (1986)zbMATHCrossRefGoogle Scholar
  84. 84.
    Leahu, H.: On the Bernstein–von Mises phenomenon in the Gaussian white noise model. Electron. J. Stat. 5, 373–404 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Lehmann, E.L.: “Student” and small-sample theory. Stat. Sci. 14(4), 418–426 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Lehmann, E.L.: Optimality and symposia: some history. Lect. Notes Monogr. Ser. 44, 1–10 (2004)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Lehmann, E.L.: Some history of optimality. Lect. Notes Monogr. Ser. 57, 11–17 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Lenhard, J.: Models and statistical inference: the controversy between Fisher and Neyman–Pearson. Br. J. Philos. Sci. 57(1), 69–91 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Leonard, R.: Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900–1960. Cambridge University Press, New York (2010)zbMATHCrossRefGoogle Scholar
  90. 90.
    Lynch, P.: The origins of computer weather prediction and climate modeling. J. Comput. Phys. 227(7), 3431–3444 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Madansky, A.: Bounds on the expectation of a convex function of a multivariate random variable. Ann. Math. Stat. 743–746 (1959)Google Scholar
  92. 92.
    Madansky, A.: Inequalities for stochastic linear programming problems. Manag. Sci. 6(2), 197–204 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Mangel, M., Samaniego, F.J.: Abraham Wald’s work on aircraft survivability. J. A. S. A. 79(386), 259–267 (1984)CrossRefGoogle Scholar
  94. 94.
    Marshall, A.W., Olkin, I.: Multivariate Chebyshev inequalities. Ann. Math. Stat. 31(4), 1001–1014 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering, vol. 143. Academic [Harcourt Brace Jovanovich Publishers], New York (1979)Google Scholar
  96. 96.
    McKerns, M.M., Strand, L., Sullivan, T.J., Fang, A., Aivazis, M.A.G.: Building a framework for predictive science. In: Proceedings of the 10th Python in Science Conference (SciPy 2011) (2011)Google Scholar
  97. 97.
    Morgenstern, O.: Abraham Wald, 1902–1950. Econometrica: J. Econom. Soci. 361–367 (1951)Google Scholar
  98. 98.
    Mulholland, H.P., Rogers, C.A.: Representation theorems for distribution functions. Proc. Lond. Math. Soc. (3) 8(2), 177–223 (1958)Google Scholar
  99. 99.
    Nash, J.: Non-cooperative games. Ann. Math. (2) 54, 286–295 (1951)Google Scholar
  100. 100.
    Nash, J.F. Jr.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. U. S. A. 36, 48–49 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Nemirovsky, A.S.: Information-based complexity of linear operator equations. J. Complex. 8(2), 153–175 (1992)MathSciNetCrossRefGoogle Scholar
  102. 102.
    Neyman, J.: Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. R. Soc. Lond. Ser. A 236(767), 333–380 (1937)zbMATHCrossRefGoogle Scholar
  103. 103.
    Neyman, J.: A Selection of Early Statistical Papers of J. Neyman. University of California Press, Berkeley (1967)Google Scholar
  104. 104.
    Neyman, J., Pearson, E.S.: On the use and interpretation of certain test criteria for purposes of statistical inference. Biometrika 20A, 175–240, 263–294 (1928)zbMATHGoogle Scholar
  105. 105.
    Neyman, J., Pearson, E.S.: On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. R. Soc. Lond. Ser. A 231, 289–337 (1933)zbMATHCrossRefGoogle Scholar
  106. 106.
    Olkin, I., Pratt, J.W.: A multivariate Tchebycheff inequality. Ann. Math. Stat. 29(1), 226–234 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    Owhadi, H.: Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games. SIAM Rev. (Research spotlights) (2016, to appear). arXiv:1503.03467Google Scholar
  108. 108.
    Owhadi, H., Scovel, C.: Qualitative robustness in Bayesian inference. arXiv:1411.3984 (2014)Google Scholar
  109. 109.
    Owhadi, H., Scovel, C.: Brittleness of Bayesian inference and new Selberg formulas. Commun. Math. Sci. 14(1), 83–145 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Owhadi, H., Scovel, C.: Extreme points of a ball about a measure with finite support. Commun. Math. Sci. (2015, to appear). arXiv:1504.06745Google Scholar
  111. 111.
    Owhadi, H., Scovel, C.: Separability of reproducing kernel Hilbert spaces. Proc. Am. Math. Soc. (2015, to appear). arXiv:1506.04288Google Scholar
  112. 112.
    Owhadi, H., Scovel, C., Sullivan, T.J.: Brittleness of Bayesian inference under finite information in a continuous world. Electron. J. Stat. 9, 1–79 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Owhadi, H., Scovel, C., Sullivan, T.J.: On the Brittleness of Bayesian Inference. SIAM Rev. (Research Spotlights) (2015)Google Scholar
  114. 114.
    Owhadi, H., Scovel, C., Sullivan, T.J., McKerns, M., Ortiz, M.: Optimal Uncertainty Quantification. SIAM Rev. 55(2), 271–345 (2013)Google Scholar
  115. 115.
    Packel, E.W.: The algorithm designer versus nature: a game-theoretic approach to information-based complexity. J. Complex. 3(3), 244–257 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Pearson, E.S.: ‘Student’ A Statistical Biography of William Sealy Gosset. Clarendon Press, Oxford (1990)zbMATHGoogle Scholar
  117. 117.
    Pfanzagl, J.: Conditional distributions as derivatives. Ann. Probab. 7(6), 1046–1050 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    Pinelis, I.: Exact inequalities for sums of asymmetric random variables, with applications. Probab. Theory Relat. Fields 139(3-4):605–635 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Pinelis, I.: On inequalities for sums of bounded random variables. J. Math. Inequal. 2(1), 1–7 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Platzman, G.W.: The ENIAC computations of 1950-gateway to numerical weather prediction. Bull. Am. Meteorol. Soc. 60, 302–312 (1979)CrossRefGoogle Scholar
  121. 121.
    Ressel, P.: Some continuity and measurability results on spaces of measures. Mathematica Scandinavica 40, 69–78 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    Rikun, A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10(4), 425–437 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12(2), 268–285 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Rojo, J.: Optimality: The Second Erich L. Lehmann Symposium. IMS, Beachwood (2006)zbMATHGoogle Scholar
  125. 125.
    Rojo, J.: Optimality: The Third Erich L. Lehmann Symposium. IMS, Beachwood (2009)zbMATHGoogle Scholar
  126. 126.
    Rojo, J., Pérez-Abreu, V.: The First Erich L. Lehmann Symposium: Optimality. IMS, Beachwood (2004)zbMATHGoogle Scholar
  127. 127.
    Rustem, B., Howe, M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton (2002)zbMATHGoogle Scholar
  128. 128.
    Savage, L.J.: The theory of statistical decision. J. Am. Stat. Assoc. 46, 55–67 (1951)zbMATHCrossRefGoogle Scholar
  129. 129.
    Scovel, C., Hush, D., Steinwart, I.: Approximate duality. J. Optim. Theory Appl. 135(3), 429–443 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    Shapiro, A., Kleywegt, A.: Minimax analysis of stochastic problems. Optim. Methods Softw. 17(3), 523–542 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22(1), 245–270 (1997)MathSciNetzbMATHGoogle Scholar
  132. 132.
    Singpurwalla, N.D., Swift, A.: Network reliability and Borel’s paradox. Am. Stat. 55(3), 213–218 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Smith, J.E.: Generalized Chebychev inequalities: theory and applications in decision analysis. Oper. Res. 43(5), 807–825 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    Sniedovich, M.: The art and science of modeling decision-making under severe uncertainty. Decis. Mak. Manuf. Serv. 1(1–2), 111–136 (2007)MathSciNetzbMATHGoogle Scholar
  135. 135.
    Sniedovich, M.: A classical decision theoretic perspective on worst-case analysis. Appl. Math. 56(5), 499–509 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    Sniedovich, M.: Black Swans, new Nostradamuses, Voodoo decision theories, and the science of decision making in the face of severe uncertainty. Int. Trans. Oper. Res. 19(1–2), 253–281 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  137. 137.
    Spanos, A.: Why the Decision-Theoretic Perspective Misrepresents Frequentist Inference (2014). https://secure.hosting.vt.edu/www.econ.vt.edu/directory/spanos/spanos10.pdf Google Scholar
  138. 138.
    Stein, C.: Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. I, pp. 197–206. University of California Press, Berkeley/Los Angeles (1956)Google Scholar
  139. 139.
    Strasser, H.: Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory, vol. 7. Walter de Gruyter, Berlin/New York (1985)zbMATHCrossRefGoogle Scholar
  140. 140.
    Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numer. 19, 451–559 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    Student: The probable error of a mean. Biometrika 1–25 (1908)Google Scholar
  142. 142.
    Sullivan, T.J., McKerns, M., Meyer, D., Theil, F., Owhadi, H., Ortiz, M.: Optimal uncertainty quantification for legacy data observations of Lipschitz functions. ESAIM Math. Model. Numer. Anal. 47(6), 1657–1689 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Tintner, G.: Abraham Wald’s contributions to econometrics. Ann. Math. Stat. 23, 21–28 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  144. 144.
    Tjur, T.: Conditional Probability Distributions, Lecture Notes, No. 2. Institute of Mathematical Statistics, University of Copenhagen, Copenhagen (1974)Google Scholar
  145. 145.
    Tjur, T.: Probability Based on Radon Measures. Wiley Series in Probability and Mathematical Statistics. Wiley, Chichester (1980)zbMATHGoogle Scholar
  146. 146.
    Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Computer Science and Scientific Computing. Academic, Boston (1988). With contributions by A. G. Werschulz and T. BoultGoogle Scholar
  147. 147.
    Tukey, J.W.: Statistical and Quantitative Methodology. Trends in Social Science, pp. 84–136. Philisophical Library, New York (1961)Google Scholar
  148. 148.
    Tukey, J.W.: The future of data analysis. Ann. Math. Stat. 33, 1–67 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  149. 149.
    Valiant, L.G.: A theory of the learnable. Commun. ACM 27(11), 1134–1142 (1984)zbMATHCrossRefGoogle Scholar
  150. 150.
    Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49(1), 52–64 (electronic) (2007)Google Scholar
  151. 151.
    Varadarajan, V.S.: Groups of automorphisms of Borel spaces. Trans. Am. Math. Soc. 109(2), 191–220 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    von Mises, R.: Mathematical Theory of Probability and Statistics. Edited and Complemented by Hilda Geiringer. Academic, New York (1964)zbMATHGoogle Scholar
  153. 153.
    Von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Math. Ann. 100(1), 295–320 (1928)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 154.
    Von Neumann, J., Goldstine, H.H.: Numerical inverting of matrices of high order. Bull. Am. Math. Soc. 53, 1021–1099 (1947)MathSciNetzbMATHCrossRefGoogle Scholar
  155. 155.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  156. 156.
    Wald, A.: Contributions to the theory of statistical estimation and testing hypotheses. Ann. Math. Stat. 10(4), 299–326 (1939)MathSciNetzbMATHCrossRefGoogle Scholar
  157. 157.
    Wald, A.: Statistical decision functions which minimize the maximum risk. Ann. Math. (2) 46, 265–280 (1945)Google Scholar
  158. 158.
    Wald, A.: An essentially complete class of admissible decision functions. Ann. Math. Stat. 18, 549–555 (1947)MathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    Wald, A.: Sequential Analysis. 1947.zbMATHGoogle Scholar
  160. 160.
    Wald, A.: Statistical decision functions. Ann. Math. Stat. 20, 165–205 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    Wald, A.: Statistical Decision Functions. Wiley, New York (1950)zbMATHGoogle Scholar
  162. 162.
    Wald, A., Wolfowitz, J.: Optimum character of the sequential probability ratio test. Ann. Math. Stat. 19(3), 326–339 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  163. 163.
    Wald, A., Wolfowitz, J.: Characterization of the minimal complete class of decision functions when the number of distributions and decisions is finite. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 149–157. University of California Press, Berkeley (1951)Google Scholar
  164. 164.
    Wasserman, L.: Rise of the Machines. Past, Present and Future of Statistical Science. CRC Press, Boca Raton (2013)Google Scholar
  165. 165.
    Wasserman, L., Lavine, M., Wolpert, R.L.: Linearization of Bayesian robustness problems. J. Stat. Plann. Inference 37(3), 307–316 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  166. 166.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  167. 167.
    Wilson, M.: How a story from World War II shapes Facebook today. IBM Watson (2012). http://www.fastcodesign.com/1671172/how-a-story-from-world-war-ii-shapes-facebook-today.
  168. 168.
    Winkler, G.: On the integral representation in convex noncompact sets of tight measures. Mathematische Zeitschrift 158(1), 71–77 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  169. 169.
    Winkler, G.: Extreme points of moment sets. Math. Oper. Res. 13(4), 581–587 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 170.
    Wolfowitz, J.: Abraham Wald, 1902–1950. Ann. Math. Stat. 23, 1–13 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  171. 171.
    Woźniakowski, H.: Probabilistic setting of information-based complexity. J. Complex. 2(3), 255–269 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  172. 172.
    Woźniakowski, H.: What is information-based complexity? In Essays on the Complexity of Continuous Problems, pp. 89–95. European Mathematical Society, Zürich (2009)Google Scholar
  173. 173.
    Wynn, H.P.: Introduction to Kiefer (1959) Optimum Experimental Designs. In Breakthroughs in Statistics, pp. 395–399. Springer, New York (1992)Google Scholar
  174. 174.
    Xu, L., Yu, B., Liu, W.: The distributionally robust optimization reformulation for stochastic complementarity problems. Abstr. Appl. Anal. 2014, Art. ID 469587, (2014)Google Scholar
  175. 175.
    Žáčková, J.: On minimax solutions of stochastic linear programming problems. Časopis Pěst. Mat. 91, 423–430 (1966)MathSciNetzbMATHGoogle Scholar
  176. 176.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall, Upper Saddle River (1996)zbMATHGoogle Scholar
  177. 177.
    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137(1-2, Ser. A), 167–198 (2013)Google Scholar

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Authors and Affiliations

  1. 1.Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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