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A Random-Discretization Based Monte Carlo Sampling Method and its Applications

Author

Listed:
  • James C. Fu

    (University of Manitoba)

  • Liqun Wang

    (University of Manitoba)

Abstract

Recently, several Monte Carlo methods, for example, Markov Chain Monte Carlo (MCMC), importance sampling and data-augmentation, have been developed for numerical sampling and integration in statistical inference, especially in Bayesian analysis. As dimension increases, problems of sampling and integration can become very difficult. In this manuscript, a simple numerical sampling based method is systematically developed, which is based on the concept of random discretization of the density function with respect to Lebesgue measure. This method requires the knowledge of the density function (up to a normalizing constant) only. In Bayesian context, this eliminates the “conjugate restriction” in choosing prior distributions, since functional forms of full conditionals of posterior distributions are not needed. Furthermore, this method is non-iterative, dimension-free, easy to implement and fast in computing time. Some benchmark examples in this area are used to check the efficiency and accuracy of the method. Numerical results demonstrate that this method performs well for all these examples, including an example of evaluating the small probability values of a high dimensional multivariate normal distribution. As a byproduct, this method also provides an easy way of computing maximum likelihood estimates and modes of posterior distributions.

Suggested Citation

  • James C. Fu & Liqun Wang, 2002. "A Random-Discretization Based Monte Carlo Sampling Method and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 4(1), pages 5-25, March.
    • Handle: RePEc:spr:metcap:v:4:y:2002:i:1:d:10.1023_a:1015790929604
    DOI: 10.1023/A:1015790929604
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    References listed on IDEAS

    as
    1. Chib, Siddhartha & Greenberg, Edward, 1996. "Markov Chain Monte Carlo Simulation Methods in Econometrics," Econometric Theory, Cambridge University Press, vol. 12(3), pages 409-431, August.
    2. Mark J. Schervish, 1984. "Multivariate Normal Probabilities with Error Bound," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(1), pages 81-94, March.
    3. Evans, Michael & Swartz, Timothy, 2000. "Approximating Integrals via Monte Carlo and Deterministic Methods," OUP Catalogue, Oxford University Press, number 9780198502784.
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    Cited by:

    1. Antonio Punzo & Alessandro Zini, 2012. "Discrete approximations of continuous and mixed measures on a compact interval," Statistical Papers, Springer, vol. 53(3), pages 563-575, August.
    2. Liqun Wang & James Fu, 2007. "A practical sampling approach for a Bayesian mixture model with unknown number of components," Statistical Papers, Springer, vol. 48(4), pages 631-653, October.
    3. Wang, Liqun & Lee, Chel Hee, 2014. "Discretization-based direct random sample generation," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1001-1010.

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