Convergence rate of Markov chain methods for genomic motif discovery

Annals of Statistics

Volumn 41, Issue 1, 2013, Pages 91-124

  Woodard, Dawn B ^a   Rosenthal, Jeffrey S ^b  

^a School of Operations Research and Industrial Engineering   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

DNA; Gibbs sampler; Multimodal; Slow mixing; Spectral gap

Indexed keywords

EID: 84879246601     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AOS1075     Document Type: Article

References (45)

1. ANDRIEU, C.,DOUCET, A. and HOLENSTEIN, R. (2010). Particle Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 72 269-342. MR2758115
2. BELLONI, A. and CHERNOZHUKOV, V. (2009). On the computational complexity of MCMC-based estimators in large samples. Ann. Statist. 37 2011-2055. MR2533478
3. BERK, R. H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37 51-58.
4. BHATNAGAR, N. and RANDALL, D. (2004). Torpid mixing of simulated tempering on the Potts model. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms 478-487. ACM, New York. MR2291087
5. BORGS, C., CHAYES, J. T., FRIEZE, A., KIM, J. H., TETALI, P., VIGODA, E. and VU, V. H. (1999). Torpid mixing of some MCMC algorithms in statistical physics. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science 218-229. IEEE, New York.
6. DEL MORAL, P., DOUCET, A. and JASRA, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411-436. MR2278333
7. DIACONIS, P. and SALOFF-COSTE, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696-730. MR1233621
8. DIACONIS, P. and SALOFF-COSTE, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750. MR1410112
9. DIACONIS, P. and STROOCK, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36-61. MR1097463
10. FORT, G., MOULINES, E., ROBERTS, G. O. and ROSENTHAL, J. S. (2003). On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40 123-146. MR1953771
11. GELMAN, A. and RUBIN, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457-472.
12. GEMAN, S. and GEMAN, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern. Anal. Mach. Intell. 6 721-741.
13. GEWEKE, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bayesian Statistics, 4 (PeñíScola, 1991) (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 169-193. Oxford Univ. Press, New York. MR1380276
14. GREEN, P. J. and RICHARDSON, S. (2002). Hidden Markov models and disease mapping. J. Amer. Statist. Assoc. 97 1055-1070. MR1951259
15. HANS, C., DOBRA, A. andWEST, M. (2007). Shotgun stochastic search for "large p" regression. J. Amer. Statist. Assoc. 102 507-516. MR2370849
16. JARNER, S. F. and HANSEN, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341-361. MR1731030
17. JENSEN, S. T., LIU, X. S., ZHOU, Q. and LIU, J. S. (2004). Computational discovery of gene regulatory binding motifs: A Bayesian perspective. Statist. Sci. 19 188-204. MR2082154
18. JOHNSON, A. A. and JONES, G. L. (2010). Gibbs sampling for a Bayesian hierarchical general linear model. Electron. J. Stat. 4 313-333. MR2645487
19. JONES, G. L. and HOBERT, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312-334. MR1888447
20. JONES, G. L. and HOBERT, J. P. (2004). Sufficient burn-in for Gibbs samplers for a hierarchical random effects model. Ann. Statist. 32 784-817. MR2060178
21. KAMATANI, K. (2011). Weak consistency of Markov chain Monte Carlo methods. Technical report. Available at http://arxiv.org/abs/1103.5679.
22. KELLIS, M., PATTERSON, N., BIRREN, B., BERGER, B. and LANDER, E. S. (2004). Methods in comparative genomics: Genome correspondence, gene identification and regulatory motif discovery. J. Comput. Biol. 11 319-355.
23. KULLBACK, S. (1959). Information Theory and Statistics. Wiley, New York. MR0103557
24. LAWRENCE, C. E., ALTSCHUL, S. F., BOGUSKI, M. S., LIU, J. S., NEUWALD, A. F. and WOOTTON, J. C. (1993). Detecting subtle sequence signals: A Gibbs sampling strategy for multiple alignment. Science 262 208-214.
25. LIANG, F. and WONG, W. H. (2000). Evolutionary Monte Carlo: Applications to Cp model sampling and change point problem. Statist. Sinica 10 317-342.
26. LIU, J. S. (1994). The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem. J. Amer. Statist. Assoc. 89 958-966. MR1294740
27. LIU, X.,BRUTLAG, D. L. and LIU, J. S. (2001). Bio Prospector: Discovering conserved DNA motifs in upstream regulatory regions of co-expressed genes. Pacific Symposium on Biocomputing 6 127-138.
28. LIU, J. S., NEUWALD, A. F. and LAWRENCE, C. E. (1995). Bayesian models for multiple local sequence alignment and Gibbs sampling strategies. J. Amer. Statist. Assoc. 90 1156-1170.
29. LIU, J. S.,WONG, W. H. and KONG, A. (1995). Covariance structure and convergence rate of the Gibbs sampler with various scans. J. Roy. Statist. Soc. Ser. B 57 157-169. MR1325382
30. MADRAS, N. and RANDALL, D. (2002).Markov chain decomposition for convergence rate analysis. Ann. Appl. Probab. 12 581-606. MR1910641
31. MADRAS, N. and ZHENG, Z. (2003). On the swapping algorithm. Random Structures Algorithms 22 66-97. MR1943860
32. MIRA, A. (2001). Ordering and improving the performance of Monte Carlo Markov chains. Statist. Sci. 16 340-350. MR1888449
33. MOSSEL, E. and VIGODA, E. (2006). Limitations of Markov chain Monte Carlo algorithms for Bayesian inference of phylogeny. Ann. Appl. Probab. 16 2215-2234. MR2288719
34. NEUWALD, A. F., LIU, J. S. and LAWRENCE, C. E. (1995). Gibbs motif sampling: Detection of bacterial outer membrane protein repeats. Protein Sci. 4 1618-1632.
35. PESKUN, P. H. (1973). Optimum Monte-Carlo sampling using Markov chains. Biometrika 60 607-612. MR0362823
36. ROBERTS, G. O. and ROSENTHAL, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20-71. MR2095565
37. ROBERTS, G. O. and SAHU, S. K. (2001). Approximate predetermined convergence properties of the Gibbs sampler. J. Comput. Graph. Statist. 10 216-229. MR1939698
38. ROSENTHAL, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558-566. MR1340509
39. ROSENTHAL, J. S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimators. Statist. Comput. 6 269-275.
40. ROTH, F. P., HUGHES, J. D., ESTEP, P. W. and CHURCH, G. M. (1998). Finding DNA regulatory motifs within unaligned noncoding sequences clustered by whole-genome mRNA quantitation. Nat. Biotechnol. 16 939-945.
41. SINCLAIR, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351-370. MR1211324
42. TIERNEY, L. (1998). A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8 1-9. MR1620401
43. WOODARD, D. B. and ROSENTHAL, J. S. (2013). Supplement to "Convergence rate of Markov chain methods for genomic motif discovery." DOI:10.1214/12-AOS1075SUPP.
44. WOODARD, D. B., SCHMIDLER, S. C. and HUBER, M. (2009a). Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions. Ann. Appl. Probab. 19 617-640. MR2521882
45. WOODARD, D. B., SCHMIDLER, S. C. and HUBER, M. (2009b). Sufficient conditions for torpid mixing of parallel and simulated tempering. Electron. J. Probab. 14 780-804. MR2495560