Stochastic approximation Monte Carlo importance sampling for approximating exact conditional probabilities

Statistics and Computing

Volumn 24, Issue 4, 2014, Pages 505-520

  Cheon, Sooyoung ^a   Liang, Faming ^b   Chen, Yuguo ^c   Yu, Kai ^d  

^a Korea University   (South Korea)

^b TEXAS A AND M UNIVERSITY   (United States)

^c UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^d NATIONAL CANCER INSTITUTE   (United States)

Author keywords

Contingency table; Exact inference; Importance sampling; MCMC; Stochastic approximation Monte Carlo

Indexed keywords

EID: 84957428406     PISSN: 09603174     EISSN: 15731375     Source Type: Journal    
DOI: 10.1007/s11222-013-9384-6     Document Type: Article

References (54)

1. 4ti2 team: 4ti2—a software package for algebraic, geometric and combinatorial problems on linear spaces (2006). Available at www.4ti2.de
2. Agresti, A.: Categorical Data Analysis, 2nd edn. Wiley, New York (2002)
3. Andrieu, C., Moulines, a., Priouret, P.: Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44, 283–312 (2005)
4. Atchadé, Y.F., Liu, J.S.: The Wang-Landau algorithm for Monte Carlo computation in general state spaces. Stat. Sin. 20, 209–233 (2010)
5. Baglivo, J., Oliver, D., Pagano, M.: Methods for the analysis of contingency tables with large and small cell counts. J. Am. Stat. Assoc. 83, 1006–1013 (1988)
6. Berg, B.A., Neuhaus, T.: Multicanonical algorithms for first order phase transitions. Phys. Lett. B 267, 249–252 (1991)
7. Bezáková, I., Bhatnagar, N., Vigoda, E.: Sampling binary contingency tables with a greedy start. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA) (2006)
8. Bezáková, I., Sinclair, A., Stefankovic, D., Vigoda, E.: Negative examples for sequential importance sampling of binary contingency tables. In: Algorithms ESA 2006. Lecture Notes in Computer Science, vol. 4168, pp. 136–147. Springer, Berlin (2007)
9. Blanchet, J.H.: Efficient importance sampling for binary contingency tables. Ann. Appl. Probab. 19, 949–982 (2009)
10. Booth, J.G., Butler, R.W.: An importance sampling algorithm for exact conditional test in log-linear models. Biometrika 86(2), 321–332 (1999)
11. Bornn, L., Jacob, P.E., Del Moral, P., Doucet, A.: An adaptive interacting Wang-Landau algorithm for automatic density exploration (2012). arXiv:1109.3829
12. Botev, Z.I., Kroese, D.P.: An efficient algorithm for rate-event probability estimation, combinatorial optimization, and counting. Methodol. Comput. Appl. Probab. 10, 471–505 (2008)
13. Caffo, B.S., Booth, J.G.: A Markov chain Monte Carlo algorithm for approximating exact conditional probabilities. J. Comput. Graph. Stat. 10(4), 730–745 (2001)
14. Chen, H.F.: Stochastic Approximation and Its Applications. Kluwer Academic, Dordrecht (2002)
15. Chen, Y., Diaconis, P., Holmes, S.P., Liu, J.S.: Sequential Monte Carlo methods for statistical analysis of table. J. Am. Stat. Assoc. 100, 109–120 (2005)
16. Chen, Y., Dinwoodie, I.H., Sullivant, S.: Sequential importance sampling for multiway tables. Ann. Stat. 34, 523–545 (2006)
17. Chopin, N., Lelievre, T., Stoltz, G.: Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors. Stat. Comput. 22, 897–916 (2012)
18. Christensen, R.: Log-liner Models and Logistic Regression, 2nd edn. Springer, New York (1997)
19. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Springer, New York (1992)
20. Dalenius, P., Reiss, R.S.: Data-swapping: a technique for disclosure control. J. Stat. Plan. Inference 6, 73–85 (1982)
21. Darve, E., Pohorille, A.: Calculating free energies using average force. J. Chem. Phys. 115, 9169–9183 (2001)
22. De Loera, J.A., Onn, S.: Markov basis of three-way tables are arbitrarily complicated. J. Symb. Comput. 41, 173–181 (2006)
23. Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. B 68, 411–436 (2006)
24. Diaconis, P., Sturmfels, B.: Algebraic algorithms for sampling from conditional distributions. Ann. Stat. 26, 363–397 (1998)
25. Dinwoodie, I.H., Chen, Y.: Sampling large tables with constraints. Stat. Sin. 21, 1591–1609 (2011)
26. Dobra, A.: Markov bases for decomposable graphical models. Bernoulli 9, 1093–1108 (2003)
27. Faraway, J.J.: Extending the Linear Model with R. Chapman & Hall, London (2006)
28. Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T., Stoltz, G.: Convergence and efficiency of the Wang-Landau algorithm (2012). arXiv:1207.6880
29. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
30. Hout, M., Duncan, O.D., Sobel, M.E.: In: Association and Heterogeneity: Structural Models of Similarities and Differences (1987)
31. Jacob, P.E., Ryder, R.J.: The Wang-Landau algorithm reaches the Flat Histogram criterion in finite time (2011). arXiv:1110.4025
32. Jerrum, M.R., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. Assoc. Comput. Mach. 51, 671–697 (2004)
33. Kuroda, M., Hashiguchi, H., Nakagawa, S.: Computing p-values in conditional independence models for a contingency table. Comput. Stat. 25, 57–70 (2010)
34. Lauritzen, S.L.: Graphical Models. Clarendon Press, Oxford (1996)
35. Liang, F.: Dynamically weighted importance sampling in Monte Carlo computation. J. Am. Stat. Assoc. 97, 807–821 (2002)
36. Liang, F.: Annealing contour Monte Carlo for structure optimization in an off-lattice protein model. J. Chem. Phys. 120, 6756–6763 (2004)
37. Liang, F.: A generalized Wang-Landau algorithm for Monte Carlo Computation. J. Am. Stat. Assoc. 100, 1311–1327 (2005)
38. Liang, F.: A theory on flat histogram Monte Carlo methods. J. Stat. Phys. 122, 511–529 (2006)
39. Liang, F.: On the use of stochastic approximation Monte Carlo for Monte Carlo integration. Stat. Probab. Lett. 79, 581–587 (2009a)
40. Liang, F.: Improving stochastic approximation Monte Carlo by kernel smoothing: theory and applications. Ann. Stat. 37, 2626–2654 (2009b)
41. Liang, F.: Trajectory averaging for stochastic approximation MCMC algorithms. Ann. Stat. 38, 2823–2856 (2010)
42. Liang, F., Liu, C., Carroll, R.: Stochastic approximation in Monte Carlo computation. J. Am. Stat. Assoc. 102(477), 305–320 (2007)
43. Long, M.A., Berry, K.J., Mielke, P.W.: Multiway contingency tables: Monte Carlo resampling probability values for the chi-squared and likelihood-ratio tests. Psychol. Rep. 107, 501–510 (2010)
44. Marsili, S., Barducci, A., Chelli, R., Procacci, P., Schettino, V.: Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations. J. Phys. Chem. B 110, 14011–14013 (2006)
45. Mehta, C., Patel, N.: A network algorithm for the exact treatment of the 2×K contingency table. Commun. Stat. 9, 649–664 (1980)
46. Mehta, C., Patel, N., Senchauduri, P.: Importance sampling for estimating exact probabilities in permutational inference. J. Am. Stat. Assoc. 83, 999–1005 (1988)
47. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)
48. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22, 400–407 (1951)
49. Roberts, G.O., Tweedie, R.L.: Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95–110 (1996)
50. Rubinstein, R.Y., Kroese, D.P.: Simulation and the Monte Carlo Method, 2nd edn. Wiley, New York (2007)
51. Song, Q., Wu, M., Liang, F.: Weak convergence rate of population stochastic approximation MCMC algorithm. J. Appl. Probab. (2011, under review)
52. Wang, F., Landau, D.P.: Efficient, multiple-range random walk algorithm to calculate density of states. Phys. Rev. Lett. 86, 2050–2053 (2001)
53. Whittaker, J.: Graphical Models in Applied Mathematical Statistics. Wiley, New York (1990)
54. Wu, M., Liang, F.: Population SAMC vs SAMC: convergence and applications to Bayesian model selection problems. J. Biom. Biostat. S1, 002 (2011)