Fixed precision MCMC estimation by median of products of averages

Journal of Applied Probability

Volumn 46, Issue 2, 2009, Pages 309-329

  Niemiro, Wojciech ^a   Pokarowski, Piotr ^b  

^a NICOLAUS COPERNICUS UNIVERSITY   (Poland)

^b UNIVERSITY OF WARSAW   (Poland)

Author keywords

Bias; Confidence estimation; Markov chain Monte Carlo; Mean square error; Rare event simulation

Indexed keywords

EID: 67949089777     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/1245676089     Document Type: Article

References (31)

1. ALDOUS. D. (1987). On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Prob. Eng. Inf. Sci. 1, 33-46.
2. ASMUSSEN, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.
3. ASMUSSEN, S. AND BINSWANGER, K. (1997). Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27, 297-318.
4. ASMUSSEN, S. AND GLYNN, P.W. (2007). Stochastic Simulation: Algorithms and Analysis (Stoch. Modelling Appl. Prob. 57). Springer, New York.
5. ASMUSSEN, S. AND KROESE, D.P. (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545-558.
6. BERNSTEIN, S.N. (1924). On a modification of Chebyshev's inequality and of the error formula of Laplace. Ann. Sci. Inst. Savantes Ukraine, Sect. Math. 1, 38-49.
7. BREMAUD, P. (1999). Markov Chains. Springer, New York.
8. DACONIS, P. AND STROOK, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1, 36-61.
9. DIACONIS, P., HOLMES, S. AND NEAL, R.M. (2000). Analysis of a nonreversible Markov chain sampler. Ann. Appl. Prob. 10, 726-752.
10. DINWOODIE, I.H. (1995). A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Prob. 5, 37-43.
11. DYER, M. AND FRIEZE, A. (1991). Computing the volume of convex bodies: a case where randomness provably helps. In Probabilistic Combinatorics and its Applications (Proc. AMS Symp, Appl, Math, 44), American Mathematical Society, Providence, RI, pp. 123-169.
12. DYER, M., GOLDBERG, L.A. AND JERRUM, M. (2006). Systematic scan for sampling colorings. Ann. Appl. Prob. 16, 185-230.
13. FERRENBERG, A.M. AND SWENDSEN, R.H. (1989). Optimized Monte Carlo data analysis. Phys. Rev. Lett. 63, 1195-1198.
14. Fill, J.A. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Prob. 1, 62-87.
15. GILLMAN, D. (1998). A Chernoff bound for random walks on expander graphs. SIAM J. Comput. 27, 1203-1220.
16. HARTINGER, J. AND KORTSCHAK, D. (2006). On the efficiency of Asmussen-Kroese-estimator and its applications to stop-loss transforms. In 6th Internal. Workshop on Rare Event Simulation (Bamberg, October 2006), pp. 162-171.
17. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. As Soc. 58, 13-30.
18. HORN, R. AND JOHNSON, C.R. (1985). Matrix Analysis. Cambridge University Press.
19. JERRUM, M. (2003). Counting, Sampling and Integrating: Algorithms and Complexity, Birkhäuser, Basel.
20. JERRUM, M. AND SINCLAIR, A. (1993). Polynomial-time approximation algorithms for the Ising model. SIAM. J. Comput. 22, 1087-1116.
21. JERRUM, M. AND SINCLAIR, A. (1996). The Markov chain Monte Carlo method: an approach to approximate counting and integration, In Approximation Algorithms for NP-hard Problems, ed. D. Hochbaum, PWS, Boston, pp. 482-520.
22. JERRUM, M., SINCLAIR, A. AND VIGODA, E. (2004). A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. Assoc. Comput. Mach. 51, 671-697.
23. JERRUM, M. R., VALIANT, L.G. AND VAZIRANI, V.V. (1986). Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43, 169-188.
24. LEZAUD, P. (1998). Chernoff-type bound for finite Markov chains. Ann. Appl. Prob. 8, 849-867.
25. NIEMIRO, W., (2008). Nonasymptotic bounds on the estimation error for regenerative MCMC algorithm under a drift condition. Submitted.
26. NEWMAN, M.E.J. AND BARKEMA, G.T. (1999). Monte Carlo Methods in Statistical Physics. Clarendon Press, New York.
27. POKAROWSKI, P., DROSTE, K. AND KOLINSKI, A. (2005). A minimal protein-like lattice model: an alpha-helix motif. J. Chem. Phys. 122, 214915.
28. POKAROWSKI, P., KOLINSKI, A., AND SKOLNICK, J. (2003). A minimal physically realistic protein-like lattice model: designing an energy landscape that ensures all-or-none folding to a unique native state. Biophysical J. 84, 1518-1526.
29. SANDMAN, W. (ed.) (2006). Proceedings of the 6th International Workshop on Rare Event Simulation (Bamberg, October 2006), University of Bamberg, Germany.
30. SINCLAIR, A.J. AND JERRUM, M.R. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. Inf. Comput. 82, 93-133.
31. SOKAL, A.D. (1989). Monte Carlo methods in statistical mechanics: foundations and new algorithm. Lecture Notes: Cours de Troisieme Cycle de la Physique en Suisse Romande (Lausanne, June 1989). Unpublished manuscript.