Statistical Science

Comment: Unreasonable Effectiveness of Monte Carlo

Art B. Owen

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Abstract

There is a role for statistical computation in numerical integration. However, the competition from incumbent methods looks to be stiffer for this problem than for some of the newer problems being handled by probabilistic numerics. One of the challenges is the unreasonable effectiveness of the central limit theorem. Another is the unreasonable effectiveness of pseudorandom number generators. A third is the common $O(n^{3})$ cost of methods based on Gaussian processes. Despite these advantages, the classical methods are weak in places where probabilistic methods could bring an improvement.

Article information

Source
Statist. Sci., Volume 34, Number 1 (2019), 29-33.

Dates
First available in Project Euclid: 12 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ss/1555056027

Digital Object Identifier
doi:10.1214/18-STS676

Mathematical Reviews number (MathSciNet)
MR3938960

Keywords
Probabilistic numerics quasi-Monte Carlo

Citation

Owen, Art B. Comment: Unreasonable Effectiveness of Monte Carlo. Statist. Sci. 34 (2019), no. 1, 29--33. doi:10.1214/18-STS676. https://projecteuclid.org/euclid.ss/1555056027


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References

  • Cockayne, J., Oates, C., Sullivan, T. and Girolami, M. (2017). Bayesian probabilistic numerical methods. Technical report. Available at arXiv:1702.03673.
  • Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.
  • Diaconis, P. (1988). Bayesian numerical analysis. In Statistical Decision Theory and Related Topics, IV, Vol. 1 (West Lafayette, Ind., 1986) 163–175. Springer, New York.
  • Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist. 14 1–67.
  • Dick, J. (2011). Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39 1372–1398.
  • Dick, J. and Pillichshammer, F. (2010). Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge Univ. Press, Cambridge.
  • Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57. CRC Press, New York.
  • Ferrenberg, A. M. Landau, D. P. and Wong, Y. J. (1992). Monte Carlo simulations: Hidden errors from “good” random number generators. Phys. Rev. Lett. 69 3382.
  • Gelman, A. and Shirley, K. (2011). Inference from simulations and monitoring convergence. In Handbook of Markov Chain Monte Carlo (S. Brooks, A. Gelman, G. Jones and X.-L. Meng, eds.). Chapman & Hall/CRC Handb. Mod. Stat. Methods 163–174. CRC Press, Boca Raton, FL.
  • Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16 927–985.
  • Hennig, P., Osborne, M. A. and Girolami, M. (2015). Probabilistic numerics and uncertainty in computations. Proceedings of the Royal Statistical Society, A 471 20150142, 17.
  • Jagadeeswaran, R. and Hickernell, F. J. (2018). Fast automatic Bayesian cubature using lattice sampling. Technical report. Available at arXiv:1809.09803.
  • L’Ecuyer, P. and Lemieux, C. (2002). A survey of randomized quasi-Monte Carlo methods. In Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications (M. Dror, P. L’Ecuyer and F. Szidarovszki, eds.) 419–474. Kluwer Academic, Boston, MA.
  • L’Ecuyer, P. and Simard, R. (2007). TestU01: A C library for empirical testing of random number generators. ACM Trans. Math. Software 33 Art. 22.
  • Lavine, M. and Hodges, J. (2019). Intuition for an old curiosity and an implication for MCMC. Amer. Statist. To appear. DOI:10.1080/00031305.2018.1518267.
  • Lewis, P. A. W., Goodman, A. S. and Miller, J. M. (1969). A pseudo-random number generator for the System/360. IBM System Journal 8 136–146.
  • Niederreiter, H. (1978). Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc. 84 957–1041.
  • Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics 63. SIAM, Philadelphia, PA.
  • Owen, A. B. (1992). Empirical likelihood and small samples. In Computing Science and Statistics 79–88. Springer, Berlin.
  • Owen, A. B. (2018). Effective dimension of some weighted pre-Sobolev spaces with dominating mixed partial derivatives. Technical report. Available at arXiv:1709.06695.
  • Peng, L. (2004). Empirical-likelihood-based confidence interval for the mean with a heavy-tailed distribution. Ann. Statist. 32 1192–1214.
  • Ritter, K. (2000). Average-Case Analysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, Berlin.
  • Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409–435.
  • Sloan, I. H. (2007). Finite-order integration weights can be dangerous. Comput. Methods Appl. Math. 7 239–254.
  • Surjanovic, S. and Bingham, D. (2014). Virtual library of simulation experiments: Test functions and datasets. Available at http://www.sfu.ca/~ssurjano.
  • Tan, Z. (2004). On a likelihood approach for Monte Carlo integration. J. Amer. Statist. Assoc. 99 1027–1036.
  • Vats, D., Flegal, J. M. and Jones, G. L. (2015). Multivariate output analysis for Markov chain Monte Carlo. Technical report. Available at arXiv:1512.07713.
  • Xu, W. and Stein, M. L. (2017). Maximum likelihood estimation for a smooth Gaussian random field model. SIAM/ASA J. Uncertain. Quantificat. 5 138–175.

See also

  • Main article: Probabilistic Integration: A Role in Statistical Computation?.