Computational Statistics

, Volume 16, Issue 1, pp 1–23 | Cite as

Sampling algorithms for estimating the mean of bounded random variables

  • Jian Cheng


We show two distribution-independent algorithms to estimate the mean of bounded random variables, one with the knowledge of variance, the other without. These algorithms guarantee that the estimate is within the desired precision with an error probability less than or equal to the requirement. Some simplified stopping rules are also given.


Algorithm Sampling Monte Carlo estimation Confidence intervals 



I would like to thank the anonymous reviewers and Marek J. Druzdzel for their useful comments that considerably improved the presentation of this paper.


  1. Bennett, G. (1962), ‘Probability inequalities for the sum of independent random variables’, Journal of the American Statistical Association 57, 33–45.CrossRefGoogle Scholar
  2. Bernstein, S. (1927), Theory of Probability, Moscow.Google Scholar
  3. Canetti, R., Even, G. & Goldreich, O. (1995), ‘Lower bounds for sampling algorithms for estimating the average’, Information Processing Letters 53, 17–25.MathSciNetCrossRefGoogle Scholar
  4. Cheng, J. & Druzdzel, M. J. (2000), ‘AIS-BN: An adaptive importance sampling algorithm for evidential reasoning in large Bayesian networks’, Journal of Artificial Intelligence Research 13, 155–188.MathSciNetCrossRefGoogle Scholar
  5. Dagum, P. & Chavez, M. R. (1993), ‘Approximating probabilistic inference in Bayesian belief networks’, IEEE Transactions on Pattern Analysis and Machine Intelligence 15(3), 246–255.CrossRefGoogle Scholar
  6. Dagum, P. & Horvitz, E. (1993), ‘A Bayesian analysis of simulation algorithms for inference in belief networks’, Networks 23, 499–516.MathSciNetCrossRefGoogle Scholar
  7. Dagum, P., Karp, R., Luby, M. & Ross, S. (2000), ‘An optimal algorithm for Monte Carlo estimation’, SIAM Journal on Computing 29(5), 1481–1496.MathSciNetCrossRefGoogle Scholar
  8. Dagum, P. & Luby, M. (1997), ‘An optimal approximation algorithm for Bayesian inference’, Artificial Intelligence 93, 1–27.MathSciNetCrossRefGoogle Scholar
  9. Hoeffding, W. (1963), ‘Probability inequalities for sums of bounded random variables’, Journal of the American Statistical Association 58, 13–29.MathSciNetCrossRefGoogle Scholar
  10. Karmarkar, N., Karp, R., Lipton, R., Lovasz, L. & Luby, M. (1993), ‘A Monte-Carlo algorithm for estimating the permanent’, SIAM Journal on Computing 22(2), 284–293.MathSciNetCrossRefGoogle Scholar
  11. Karp, R. M., Luby, M. & Madras, N. (1989), ‘Monte Carlo approximation algorithms for enumeration problems’, Journal of Algorithms 10, 429–448.MathSciNetCrossRefGoogle Scholar
  12. Pradhan, M. & Dagum, P. (1996), Optimal Monte Carlo estimation of belief networks inference, in ‘Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence (UAI-96)’, Portland, Oregon, pp. 446–453.Google Scholar
  13. Siegmund, D. (1985), Sequential Analysis: tests and confidence intervals, Springer-Verlag, New York.CrossRefGoogle Scholar

Copyright information

© Physica-Verlag 2001

Authors and Affiliations

  • Jian Cheng
    • 1
  1. 1.Decision Systems Laboratory, School of Information SciencesUniversity of PittsburghPittsburghUSA

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