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Bayesian nonparametric analysis of reversible Markov chains. (English) Zbl 1360.62481
Summary: We introduce a three-parameter random walk with reinforcement, called the \((\theta,\alpha,\beta)\) scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter \(\beta\) smoothly tunes the \((\theta,\alpha,\beta)\) scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters \(\alpha\) and \(\theta\) modulate how many states are typically visited. Resorting to de Finetti’s theorem for Markov chains, we use the \((\theta,\alpha,\beta)\) scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in Bayesian nonparametric inference for species sampling problems with data generated from a reversible Markov chain with an unknown transition kernel. As a real example, we analyze data from molecular dynamics simulations of protein folding.

62M99 Inference from stochastic processes
62M02 Markov processes: hypothesis testing
62C10 Bayesian problems; characterization of Bayes procedures
60G50 Sums of independent random variables; random walks
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[1] Bacallado, S. (2011). Bayesian analysis of variable-order, reversible Markov chains. Ann. Statist. 39 838-864. · Zbl 1215.62083 · doi:10.1214/10-AOS857
[2] Bacallado, S., Favaro, S. and Trippa, L. (2013). Supplement to “Bayesian nonparametric analysis of reversible Markov chains.” . · Zbl 1360.62481 · dx.doi.org
[3] Beal, M. J., Ghahramani, Z. and Rasmussen, C. E. (2002). The infinite hidden Markov model. Adv. Neural Inf. Process. Syst. 14 577-584.
[4] Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353-355. · Zbl 0276.62010 · doi:10.1214/aos/1176342372
[5] Bunge, J. and Fitzpatrick, M. (1993). Estimating the number of species: A review. J. Amer. Statist. Assoc. 88 364-373.
[6] Comtet, L. (1974). Advanced Combinatorics : The Art of Finite and Infinite Expansions , enlarged ed. Reidel, Dordrecht. · Zbl 0283.05001
[7] Diaconis, P. (1988). Recent progress on de Finetti notions of exchangeability. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 111-125. Oxford Univ. Press, New York. · Zbl 0707.60033
[8] Diaconis, P. and Freedman, D. (1980). de Finetti’s theorem for Markov chains. Ann. Probab. 8 115-130. · Zbl 0426.60064 · doi:10.1214/aop/1176994828
[9] Diaconis, P. and Rolles, S. W. W. (2006). Bayesian analysis for reversible Markov chains. Ann. Statist. 34 1270-1292. · Zbl 1118.62085 · doi:10.1214/009053606000000290
[10] Engen, S. (1978). Stochastic Abundance Models : With Emphasis on Biological Communities and Species Diversity . Chapman & Hall, London. · Zbl 0429.62075
[11] Favaro, S., Lijoi, A., Mena, R. H. and Prünster, I. (2009). Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 993-1008. · doi:10.1111/j.1467-9868.2009.00717.x
[12] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037 · doi:10.1214/aos/1176342360
[13] Fortini, S. and Petrone, S. (2012). Hierarchical reinforced urn processes. Statist. Probab. Lett. 82 1521-1529. · Zbl 1282.62018
[14] Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161-173. · Zbl 1014.62006 · doi:10.1198/016214501750332758
[15] Keane, M. S. and Rolles, S. W. W. (2000). Edge-reinforced random walk on finite graphs. In Infinite Dimensional Stochastic Analysis ( Amsterdam , 1999). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52 217-234. R. Neth. Acad. Arts Sci., Amsterdam. · Zbl 0986.05092
[16] Lijoi, A., Mena, R. H. and Prünster, I. (2007). Bayesian nonparametric estimation of the probability of discovering new species. Biometrika 94 769-786. · Zbl 1156.62374 · doi:10.1093/biomet/asm061
[17] Lijoi, A., Mena, R. H. and Prünster, I. (2007). A Bayesian nonparametric method for prediction in EST analysis. BMC Bioinformatics 8 339-349.
[18] Merkl, F. and Rolles, S. W. W. (2009). Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 1679-1714. · Zbl 1180.82085 · doi:10.1214/08-AOP446
[19] Pande, V. S., Beauchamp, K. and Bowman, G. R. (2010). Everything you wanted to know about Markov State Models but were afraid to ask. Methods 52 99-105.
[20] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics , Probability and Game Theory. Institute of Mathematical Statistics Lecture Notes-Monograph Series (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.) 30 245-267. IMS, Hayward, CA. · doi:10.1214/lnms/1215453576
[21] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900. · Zbl 0880.60076 · doi:10.1214/aop/1024404422
[22] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. In Proceedings of the Seventh International Conference on Random Structures and Algorithms ( Atlanta , GA , 1995) 9 223-252. Wiley, New York. · Zbl 0859.60067 · doi:10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O
[23] Rolles, S. W. W. (2003). How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 243-260. · Zbl 1029.60089 · doi:10.1007/s00440-003-0260-8
[24] Shaw, D. E. (2010). Atomic-level characterization of the structural dynamics of proteins. Science 330 341-346.
[25] Teh, Y. W. and Jordan, M. I. (2010). Hierarchical Bayesian nonparametric models with applications. In Bayesian Nonparametrics 158-207. Cambridge Univ. Press, Cambridge. · doi:10.1017/CBO9780511802478.006
[26] Teh, Y. W., Jordan, M. I., Beal, M. J. and Blei, D. M. (2006). Hierarchical Dirichlet processes. J. Amer. Statist. Assoc. 101 1566-1581. · Zbl 1171.62349 · doi:10.1198/016214506000000302
[27] Zabell, S. L. (1982). W. E. Johnson’s “sufficientness” postulate. Ann. Statist. 10 1090-1099 (1 plate). · Zbl 0512.62007 · doi:10.1214/aos/1176345975
[28] Zabell, S. L. (2005). The continuum of inductive methods revisited. In Symmetry and its Discontents : Essays on the History of Inductive Probability . Cambridge Univ. Press, New York. · Zbl 1100.01001
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