Algorithmica

, Volume 64, Issue 4, pp 606–620 | Cite as

Negative Examples for Sequential Importance Sampling of Binary Contingency Tables

  • Ivona Bezáková
  • Alistair Sinclair
  • Daniel Štefankovič
  • Eric Vigoda
Article

Abstract

The sequential importance sampling (SIS) algorithm has gained considerable popularity for its empirical success. One of its noted applications is to the binary contingency tables problem, an important problem in statistics, where the goal is to estimate the number of 0/1 matrices with prescribed row and column sums. We give a family of examples in which the SIS procedure, if run for any subexponential number of trials, will underestimate the number of tables by an exponential factor. This result holds for any of the usual design choices in the SIS algorithm, namely the ordering of the columns and rows. These are apparently the first theoretical results on the efficiency of the SIS algorithm for binary contingency tables. Finally, we present experimental evidence that the SIS algorithm is efficient for row and column sums that are regular. Our work is a first step in determining the class of inputs for which SIS is effective.

Keywords

Sequential Monte Carlo Markov chain Monte Carlo Graphs with prescribed degree sequence Zero-one table 

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References

  1. 1.
    Bayati, M., Kim, J.H., Saberi, A.: A sequential algorithm for generating random graphs. Algorithmica 58(4), 860–910 (2010) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Besag, J., Clifford P.: Sequential Monte Carlo p-values. Biometrika 78(2), 301–304 (1991) MathSciNetGoogle Scholar
  3. 3.
    Bezáková, I., Bhatnagar, N., Vigoda, E.: Sampling binary contingency tables with a greedy start. Random Struct. Algorithms 30(1–2), 168–205 (2007) MATHCrossRefGoogle Scholar
  4. 4.
    Bezáková, I., Sinclair, A., Štefankovič, D., Vigoda, E.: Negative examples for sequential importance sampling of binary contingency tables. Version available on the arXiv at: http://arxiv.org/abs/math/0606650
  5. 5.
    Blanchet, J.H.: Efficient importance sampling for binary contingency tables. Ann. Appl. Probab. 19(3), 949–982 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6(4), 489–522 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chen, Y., Diaconis, P., Holmes, S.P., Liu, J.S.: Sequential Monte Carlo methods for statistical analysis of tables. J. Am. Stat. Assoc. 100(469), 109–120 (2005) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chen, Y., Dinwoodie, I.H., Sullivant, S.: Sequential importance sampling for multiway tables. Ann. Stat. 34(1), 523–545 (2006) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    De Iorio, M., Griffiths, R.C., Leblois, R., Rousset, F.: Stepwise mutation likelihood computation by sequential importance sampling in subdivided population models. Theor. Popul. Biol. 68, 41–53 (2005) MATHCrossRefGoogle Scholar
  10. 10.
    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, New York (2009) MATHCrossRefGoogle Scholar
  11. 11.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. Assoc. Comput. Mach. 51(4), 671–697 (2004) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Míguez, J., Djurić, P.M.: Blind equalization by sequential importance sampling. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 845–848 (2002) Google Scholar
  13. 13.
    Tutte, W.T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Zhang, J.L., Liu, J.S.: A new sequential importance sampling method and its application to the two-dimensional hydrophobic-hydrophilic model. J. Chem. Phys. 117(7), 3492–3498 (2002) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Ivona Bezáková
    • 1
  • Alistair Sinclair
    • 2
  • Daniel Štefankovič
    • 3
  • Eric Vigoda
    • 4
  1. 1.Department of Computer ScienceRochester Institute of TechnologyRochesterUSA
  2. 2.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of Computer ScienceUniversity of RochesterRochesterUSA
  4. 4.College of ComputingGeorgia Institute of TechnologyAtlantaUSA

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