• Luca Martino
  • David Luengo
  • Joaquín Míguez
Part of the Statistics and Computing book series (SCO)


This chapter provides an introduction to the different approaches available for sampling from a given probability distribution. We start with a brief history of the Monte Carlo (MC) method, one of the most influential algorithms of the twentieth century and the main driver for the current widespread use of random samples in many scientific fields. Then we discuss the need for MC approaches through a few selected examples, starting with two important classical applications (numerical integration and importance sampling), and finishing with two more recent developments (inverse Monte Carlo and quasi Monte Carlo). This is followed by a review of the three types of “random numbers” which can be generated (“truly” random, pseudo-random, and quasi-random), a brief description of some pseudo-random number generators and an overview of the different classes of random sampling methods available in the literature: direct, accept/reject, MCMC, importance sampling, and hybrid. Finally, the chapter concludes with an exposition of the motivation, goals, and organization of the book.


  1. 1.
    K. Alligood, T. Sauer, J.A. York, Chaos: An Introduction to Dynamical Systems (Springer, New York, 1997)CrossRefGoogle Scholar
  2. 2.
    C. Andrieu, N. de Freitas, A. Doucet, Sequential MCMC for Bayesian model selection, in Proceedings of the IEEE HOS Workshop (1999)Google Scholar
  3. 3.
    C. Andrieu, A. Doucet, R. Holenstein, Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. B 72(3), 269–342 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    M.J. Appel, R. Labarre, D. Radulovic, On accelerated random search. SIAM J. Optim. 14(3), 708–730 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M.S. Arulumpalam, S. Maskell, N. Gordon, T. Klapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50(2), 174–188 (2002)CrossRefGoogle Scholar
  6. 6.
    L. Badger, Lazzarini’s lucky approximation of π. Math. Mag. 67(2), 83–91 (1994)Google Scholar
  7. 7.
    R. Bellazzi, P. Magni, G. De Nicolao, Bayesian analysis of blood glucose time series from diabetes home monitoring. IEEE Trans. Biomed. Eng. 47(7), 971–975 (2000)CrossRefGoogle Scholar
  8. 8.
    C. Berzuini, W. Gilks, Resample-move filtering with cross-model jumps, in Sequential Monte Carlo Methods in Practice, ed. by A. Doucet, N. de Freitas, N. Gordon, Chap. 6 (Springer, New York, 2001)Google Scholar
  9. 9.
    C. Berzuini, N.G. Best, W. Gilks, C. Larizza, Dynamic conditional independence models and Markov chain Monte Carlo methods. J. Am. Stat. Assoc. 92, 1403–1412 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    A. Beskos, D. Crisan, A. Jasra, On the stability of sequential Monte Carlo methods in high dimensions. Ann. Appl. Probab. 24(4), 1396–1445 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    E. Bolviken, G. Storvik, Deterministic and stochastic particle filters in state-space models, in Sequential Monte Carlo Methods in Practice, ed. by A. Doucet, N. de Freitas, N. Gordon, Chap. 5 (Springer, New York, 2001), pp. 97–116Google Scholar
  12. 12.
    E. Bolviken, P.J. Acklam, N. Christophersen, J.M. Stordal, Monte Carlo filters for non-linear state estimation. Automatica 37(2), 177–183 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A. Boyarsky, P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Birkhäuser, Boston, 1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    R.P. Brent, Uniform random number generators for supercomputers, in Proceedings of the 5th Australian Supercomputer Conference, Melbourne (1992), pp. 95–104Google Scholar
  15. 15.
    B.S. Caffo, J.G. Booth, A.C. Davison, Empirical supremum rejection sampling. Biometrika 89(4), 745–754 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    J. Candy, Bayesian Signal Processing: Classical, Modern and Particle Filtering Methods (Wiley, Hoboken, 2009)CrossRefGoogle Scholar
  17. 17.
    O. Cappé, A. Gullin, J.M. Marin, C.P. Robert, Population Monte Carlo. J. Comput. Graph. Stat. 13(4), 907–929 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Casella, C.P. Robert, Rao-Blackwellisation of sampling schemes. Biometrika 83(1), 81–94 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    G. Casella, C.P. Robert, Post-processing accept-reject samples: recycling and rescaling. J. Comput. Graph. Stat. 7(2), 139–157 (1998)MathSciNetGoogle Scholar
  20. 20.
    G. Chaitin, On the length of programs for computing finite binary sequences. J. ACM 13, 547–569 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G. Chaitin, On the length of programs for computing finite binary sequences: statistical considerations. J. ACM 16, 145–159 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    R. Chen, Another look at rejection sampling through importance sampling. Stat. Probab. Lett. 72, 277–283 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    R. Chen, J.S. Liu, Mixture Kalman filters. J. R. Stat. Soc. B 62, 493–508 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    J. Dagpunar, Principles of Random Variate Generation (Clarendon Press, Oxford/New York, 1988)zbMATHGoogle Scholar
  25. 25.
    M.H. DeGroot, M.J. Schervish, Probability and Statistics, 3rd edn. (Addison-Wesley, New York, 2002)Google Scholar
  26. 26.
    P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications (Springer, New York, 2004)zbMATHCrossRefGoogle Scholar
  27. 27.
    P. Del Moral, A. Doucet, A. Jasra, Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68(3), 411–436 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    L. Devroye, Random variate generation for unimodal and monotone densities. Computing 32, 43–68 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    P.M. Djurić, S.J. Godsill (eds.), Special issue on Monte Carlo methods for statistical signal processing. IEEE Trans. Signal Process. 50(3), 173 (2002)Google Scholar
  30. 30.
    A. Doucet, N. de Freitas, N. Gordon (eds.), Sequential Monte Carlo Methods in Practice (Springer, New York, 2001)zbMATHGoogle Scholar
  31. 31.
    S. Duane, A.D. Kennedy, B.J. Pendleton, D. Roweth, Hybrid Monte Carlo. Phys. Lett. B 195(2), 216–222 (1987)Google Scholar
  32. 32.
    W.L. Dunn, J.K. Shultis, Exploring Monte Carlo Methods (Elsevier, Amsterdam, 2011)zbMATHGoogle Scholar
  33. 33.
    R. Eckhardt, Stan Ulam, John von Neumann, and the Monte Carlo method. Los Alamos Sci. 15, 131–137 (1987). Special Issue: Stanislaw Ulam 1909–1984Google Scholar
  34. 34.
    I. Elishakoff, Notes on philosophy of the Monte Carlo method. Int. Appl. Mech. 39(7), 753–764 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    V. Elvira, L. Martino, D. Luengo, M. Bugallo, Efficient multiple importance sampling estimators. IEEE Signal Process. Lett. 22(10), 1757–1761 (2015)CrossRefGoogle Scholar
  36. 36.
    V. Elvira, L. Martino, D. Luengo, M.F. Bugallo, Generalized multiple importance sampling (2015). arXiv:1511.03095Google Scholar
  37. 37.
    P. Fearnhead, Sequential Monte Carlo methods in Filter Theory. Ph.D. Thesis, Merton College, University of Oxford (1998)Google Scholar
  38. 38.
    Y. Fong, J. Wakefield, K. Rice, An efficient Markov chain Monte Carlo method for mixture models by neighborhood pruning. J. Comput. Graph. Stat. 21, 197–216 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, San Diego, 1996)zbMATHGoogle Scholar
  40. 40.
    J.E. Gentle, Random Number Generation and Monte Carlo Methods (Springer, New York, 2004)zbMATHGoogle Scholar
  41. 41.
    M. Gerber, N. Chopin, Sequential quasi Monte Carlo. J. R. Stat. Soc. Ser. B Stat. Methodol. 77(3), 509–579 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    C.J. Geyer, Markov Chain Monte Carlo maximum likelihood, in Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface (1991), pp. 156–163Google Scholar
  43. 43.
    W.R. Gilks, N.G.O. Robert, E.I. George, Adaptive direction sampling. Statistician 43(1), 179–189 (1994)CrossRefGoogle Scholar
  44. 44.
    W.R. Gilks, P. Wild, Adaptive rejection sampling for Gibbs sampling. Appl. Stat. 41(2), 337–348 (1992)zbMATHCrossRefGoogle Scholar
  45. 45.
    W.R. Gilks, S. Richardson, D. Spiegelhalter, Markov Chain Monte Carlo in Practice: Interdisciplinary Statistics (Taylor & Francis, London, 1995)zbMATHGoogle Scholar
  46. 46.
    B.V. Gnedenko, The Theory of Probability, 6th ed. (Gordon and Breach, Amsterdam, 1997)zbMATHGoogle Scholar
  47. 47.
    J. Goodman, A.D. Sokal, Multigrid Monte Carlo method for lattice field theories. Phys. Rev. Lett. 56(10), 1015–1018 (1986)CrossRefGoogle Scholar
  48. 48.
    N. Gordon, D. Salmond, A.F.M. Smith, Novel approach to nonlinear and non-Gaussian Bayesian state estimation. IEE Proc. F Radar Signal Process. 140, 107–113 (1993)CrossRefGoogle Scholar
  49. 49.
    P.J. Green, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711–732 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    R.C. Griths, S. Tavaré, Monte Carlo inference methods in population genetics. Math. Comput. Model. 23(8–9), 141–158 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    A. Hall, On an experimental determination of Pi. J. Messenger Math. 2, 113–114 (1873)Google Scholar
  52. 52.
    J.M. Hammersley, K.W. Morton, Poor man’s Monte Carlo. J. R. Stat. Soc. Ser. B Methodol. 16(1), 23–38 (1954)MathSciNetzbMATHGoogle Scholar
  53. 53.
    W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    W. Hörmann, J. Leydold, G. Derflinger, Automatic Nonuniform Random Variate Generation (Springer, New York, 2003)zbMATHGoogle Scholar
  55. 55.
    K. Hukushima, K. Nemoto, Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65, 1604–1608 (1996)CrossRefGoogle Scholar
  56. 56.
    C.C. Hurd, A note on early Monte Carlo computations and scientific meetings. Ann. Hist. Comput. 7(2), 141–155 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    M. Hürzeler, H.R. Künsch, Monte Carlo approximations for general state-space models. J. Comput. Graph. Stat. 7(2), 175–193 (1998)MathSciNetzbMATHGoogle Scholar
  58. 58.
    P. Jaeckel, Monte Carlo Methods in Finance (Wiley, New York, 2002)Google Scholar
  59. 59.
    A. Jasra, D.A. Stephens, C.C. Holmes, Population-based reversible jump Markov chain Monte Carlo. Biometrika 94(4), 787–807 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    L. Jing, P. Vadakkepat, Interacting MCMC particle filter for tracking maneuvering target. Digit. Signal Process. 20, 561–574 (2010)CrossRefGoogle Scholar
  61. 61.
    S. Karlin, H.M. Taylor, A First Course on Stochastic Processes (Academic, New York, 1975)zbMATHGoogle Scholar
  62. 62.
    S.K. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi, Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    D.E. Knuth, The Art of Computer Programming. Volume 2: Seminumerical Algorithms, 2nd edn. (Addison-Wesley, Reading, MA, 1981)Google Scholar
  64. 64.
    J. Kohlas, Monte Carlo Simulation in Operations Research (Springer, Berlin, 1972)zbMATHCrossRefGoogle Scholar
  65. 65.
    A.N. Kolmogorov, On tables of random numbers. Sankhya Indian J. Stat. Ser. A 25, 369–376 (1963)MathSciNetzbMATHGoogle Scholar
  66. 66.
    A.N. Kolmogorov, Three approaches to the quantitative definition of information. Probl. Inf. Transm. 1(1), 1–7 (1965)MathSciNetzbMATHGoogle Scholar
  67. 67.
    J. Kotecha, P.M. Djurić, Gaussian sum particle filtering. IEEE Trans. Signal Process. 51(10), 2602–2612 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    H.R. Künsch, Recursive Monte Carlo filters: algorithms and theoretical bounds. Ann. Stat. 33(5), 1983–2021 (2005)zbMATHCrossRefGoogle Scholar
  69. 69.
    P.K. Kythe, M.R. Schaferkotter, Handbook of Computational Methods for Integration (Chapman and Hall/CRC, Boca Raton, 2004)zbMATHCrossRefGoogle Scholar
  70. 70.
    P.-S. Laplace, Théorie Analytique des Probabilités (Mme Ve Courcier, Paris, 1812)zbMATHGoogle Scholar
  71. 71.
    A. Lasota, M.C. Mackey, Chaos, Fractals and Noise: Stochastic Aspects of Dynamics, 2nd edn. (Springer, New York, NY, 1994)zbMATHCrossRefGoogle Scholar
  72. 72.
    M. Lazzarini, Un’ applicazione del calcolo della probabilit\(\grave {\mathrm{a}}\) alla ricerca sperimentale di un valore approssimato di π. Periodico di Matematica 4, 140–143 (1901)Google Scholar
  73. 73.
    G.-L. Leclerc (Comte Buffon), Essai d’arithmétique morale. Supplément à l’Histoire Naturelle, 4 (1777)Google Scholar
  74. 74.
    D.H. Lehmer, Mathematical methods in large-scale computing units. Ann. Comput. Lab. Harv. Univ. 26, 141–146 (1951)MathSciNetzbMATHGoogle Scholar
  75. 75.
    F. Liang, C. Liu, R. Caroll, Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley Series in Computational Statistics (Wiley, London, 2010)Google Scholar
  76. 76.
    J.S. Liu, Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Stat. Comput. 6(2), 113–119 (1996)MathSciNetCrossRefGoogle Scholar
  77. 77.
    J.S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, New York, 2004)CrossRefGoogle Scholar
  78. 78.
    J.S. Liu, R. Chen, Sequential Monte Carlo methods for dynamic systems. J. Am. Stat. Assoc. 93(443), 1032–1044 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    J.S. Liu, R. Chen, W.H. Wong, Rejection control and sequential importance sampling. J. Am. Stat. Assoc. 93(443), 1022–1031 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    J.S. Liu, F. Liang, W.H. Wong, The multiple-try method and local optimization in Metropolis sampling. J. Am. Stat. Assoc. 95(449), 121–134 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    M. Ljungberg, S.E. Strand, M.A. King, Monte Carlo Calculations in Nuclear Medicine (Taylor & Francis, Boca Raton, 1998)Google Scholar
  82. 82.
    M. Locatelli, Convergence of a simulated annealing algorithm for continuous global optimization. J. Glob. Optim. 18, 219–234 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    T.R. Malthus, An Essay On The Principle Of Population (Electronic Scholarly Publishing Project, London, 1998)Google Scholar
  84. 84.
    E. Marinari, G. Parisi, Simulated tempering: a new Monte Carlo scheme. Europhys. Lett. 19(6), 451–458 (1992)CrossRefGoogle Scholar
  85. 85.
    A. Marshall, The use of multistage sampling schemes in Monte Carlo computations, in Symposium on Monte Carlo (Wiley, New York, 1956), pp. 123–140Google Scholar
  86. 86.
    P. Martin-Iöf, Complexity of oscillations in infinite binary sequences. Z. Wahrscheinlichkeitstheorie verw. Geb. 19, 225–230 (1971)MathSciNetCrossRefGoogle Scholar
  87. 87.
    R.M. May, Biological populations with nonoverlapping generations: stable points, stable cycles and chaos. Science 186, 645–647 (1974)CrossRefGoogle Scholar
  88. 88.
    R.M. May, Simple mathematical models with very complicated dynamics. Nature 261(5560), 459–467 (1976)zbMATHCrossRefGoogle Scholar
  89. 89.
    N. Metropolis, The beginning of the Monte Carlo method. Los Alamos Sci. 15, 125–130 (1987). Special Issue: Stanislaw Ulam 1909–1984Google Scholar
  90. 90.
    N. Metropolis, S. Ulam, The Monte Carlo method. J. Am. Stat. Assoc. 44, 335–341 (1949)zbMATHCrossRefGoogle Scholar
  91. 91.
    N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)CrossRefGoogle Scholar
  92. 92.
    H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (Society for Industrial and Applied Mathematics, Philadelphia, 1992)zbMATHCrossRefGoogle Scholar
  93. 93.
    A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill Series in Electrical Engineering (McGraw-Hill, New York, 1984)Google Scholar
  94. 94.
    S.K. Park, K.W. Miller, Random number generators: good ones are hard to find. Commun. ACM 31(10), 1192–1201 (1988)MathSciNetCrossRefGoogle Scholar
  95. 95.
    M.M. Pieri, H. Martel, C. Grenón, Anisotropic galactic outflows and enrichment of the intergalactic Medium. I. Monte Carlo simulations. Astrophys. J. 658(1), 36–51 (2007)CrossRefGoogle Scholar
  96. 96.
    S.B. Pope, A Monte Carlo method for the PDF equations of turbolent reactive flow. Combust. Sci. Technol. 25, 159–174 (1981)CrossRefGoogle Scholar
  97. 97.
    D. Remondo, R. Srinivasan, V.F. Nicola, W.C. van Etten, H.E.P. Tattje, Adaptive importance sampling for performance evaluation and parameter optimization of communication systems. IEEE Trans. Commun. 48(4), 557–565 (2000)CrossRefGoogle Scholar
  98. 98.
    B. Ristic, S. Arulampalam, N. Gordon, Beyond the Kalman Filter (Artech House, Boston, 2004)zbMATHGoogle Scholar
  99. 99.
    C.P. Robert, G. Casella, Monte Carlo Statistical Methods (Springer, New York, 2004)zbMATHCrossRefGoogle Scholar
  100. 100.
    M. Rosenbluth, A. Rosenbluth, Monte Carlo calculation of average extension of molecular chains. J. Chem. Phys. 23, 356–359 (1955)CrossRefGoogle Scholar
  101. 101.
    D.B. Rubin, A noniterative sampling/importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest: the SIR algorithm. J. Am. Stat. Assoc. 82, 543–546 (1987)Google Scholar
  102. 102.
    E. Segré, From X-Rays to Quarks: Modern Physicists and Their Discoveries (Freeman, New York, 1980)Google Scholar
  103. 103.
    J.I. Siepmann, A method for the direct calculation of chemical potentials for dense chain systems. Mol. Phys. 70(6), 1145–1158 (1990)CrossRefGoogle Scholar
  104. 104.
    J.I. Siepmann, D. Frenkel, Configurational bias Monte Carlo: a new sampling scheme for flexible chains. Mol. Phys. 75(1), 59–70 (1992)CrossRefGoogle Scholar
  105. 105.
    T. Siiskonen, R. Pollanen, Alpha-electron and alpha-photon coincidences in high-resolution alpha spectrometry. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 558(2), 437–440 (2006)CrossRefGoogle Scholar
  106. 106.
    R.H. Swendsen, J.S. Wang, Replica Monte Carlo simulation of spin glasses. Phys. Rev. Lett. 57(21), 2607–2609 (1986)MathSciNetCrossRefGoogle Scholar
  107. 107.
    H. Tanizaki, On the nonlinear and non-normal filter using rejection sampling. IEEE Trans. Autom. Control 44(3), 314–319 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    H. Tanizaki, Nonlinear and non-Gaussian state space modeling using sampling techniques. Ann. Inst. Stat. Math. 53(1), 63–81 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    M.D. Troutt, W.K. Pang, S.H. Hou, Vertical Density Representation and Its Applications (World Scientific, Singapore, 2004)zbMATHCrossRefGoogle Scholar
  110. 110.
    J.P. Valleau, Density-scaling: a new Monte Carlo technique in statistical mechanics. J. Comput. Phys. 96(1), 193–216 (1991)CrossRefGoogle Scholar
  111. 111.
    P.F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique 10, 113–121 (1838)Google Scholar
  112. 112.
    P.F. Verhulst, Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles 18, 1–42 (1845)Google Scholar
  113. 113.
    J. von Neumann, Various techniques in connection with random digits, in Monte Carlo Methods, ed. by A.S. Householder, G.E. Forsythe, H.H. Germond. National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, DC, 1951), pp. 36–38Google Scholar
  114. 114.
    X. Wang, Improving the rejection sampling method in quasi-Monte Carlo methods. J. Comput. Appl. Math. 114(2), 231–246 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    T. Warnock, Random-number generators. Los Alamos Sci. 15, 137–141 (1987). Special Issue: Stanislaw Ulam 1909–1984Google Scholar
  116. 116.
    E.M. Wijsman, Monte Carlo Markov chain methods and model selection in genetic epidemiology. Comput. Stat. Data Anal. 32(3–4), 349–360 (2000)zbMATHCrossRefGoogle Scholar
  117. 117.
    D. Williams, Probability with Martingales (Cambridge University Press, Cambridge, 1991)zbMATHCrossRefGoogle Scholar
  118. 118.
    S.R. Williams, D.J. Evans, Nonequilibrium dynamics and umbrella sampling. Phys. Rev. Lett. 105(11), 1–26 (2010)CrossRefGoogle Scholar
  119. 119.
    P. Zanetti, New Monte Carlo scheme for simulating Lagrangian particle diffusion with wind shear effects. Appl. Math. Model. 8(3), 188–192 (1984)MathSciNetCrossRefGoogle Scholar
  120. 120.
    P. Zhang, Nonparametric importance sampling. J. Am. Stat. Assoc. 91(435), 1245–1253 (1996)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Luca Martino
    • 1
  • David Luengo
    • 2
  • Joaquín Míguez
    • 1
  1. 1.Department of Signal Theory and CommunicationsCarlos III University of MadridMadridSpain
  2. 2.Department of Signal Theory and CommunicationsTechnical University of MadridMadridSpain

Personalised recommendations