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Why simple quadrature is just as good as Monte Carlo

Author

Listed:
  • Vanslette Kevin

    (Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA)

  • Al Alsheikh Abdullatif

    (King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia)

  • Youcef-Toumi Kamal

    (Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA)

Abstract

We motive and calculate Newton–Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton–Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor ∝N-12{\propto N^{-\frac{1}{2}}}. This dimension independent factor is validated in our simulations.

Suggested Citation

  • Vanslette Kevin & Al Alsheikh Abdullatif & Youcef-Toumi Kamal, 2020. "Why simple quadrature is just as good as Monte Carlo," Monte Carlo Methods and Applications, De Gruyter, vol. 26(1), pages 1-16, March.
    • Handle: RePEc:bpj:mcmeap:v:26:y:2020:i:1:p:1-16:n:2
    DOI: 10.1515/mcma-2020-2055
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    References listed on IDEAS

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    1. Carpenter, Bob & Gelman, Andrew & Hoffman, Matthew D. & Lee, Daniel & Goodrich, Ben & Betancourt, Michael & Brubaker, Marcus & Guo, Jiqiang & Li, Peter & Riddell, Allen, 2017. "Stan: A Probabilistic Programming Language," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 76(i01).
    2. W. R. Gilks & P. Wild, 1992. "Adaptive Rejection Sampling for Gibbs Sampling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 41(2), pages 337-348, June.
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