Gibbs sampling, conjugate priors and coupling

Sankhya: The Indian Journal of Statistics

Volumn 72, Issue 1, 2010, Pages 136-169

  Diaconis, Persi ^a   Khare, Kshitij ^b   Saloff Coste, Laurent ^c  

^a STANFORD UNIVERSITY   (United States)

^b University of Florida ^*  (United States)

^c Department of Obstetrics Gynecology   (United States)

Author keywords

Coupling; Exponential families; Gibbs sampling; Location families; Stochastic monotonicity

Indexed keywords

EID: 77953060646     PISSN: 09727671     EISSN: None     Source Type: Journal    
DOI: 10.1007/s13171-010-0004-7     Document Type: Article

References (38)

1. Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Amer. Math. Monthly, 93, 333-348.
2. Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. in Appl. Math., 8, 69-97.
3. Andersen, H. and Diaconis, P. (2007). Hit and run as a unifying device. J. Soc. Fr. Stat. & Rev. Stat. Appl., 148, 5-28.
4. Athreya, K., Doss, H. and Sethuraman, J. (1996). On the convergence of the Markov chain simulation method. Ann. Statist., 24, 89-100.
5. Barankin, E.W. and Maitra, A.P. (1963). Generalization of the Fisher-Darmois-Koopman-Pitman theorem on sufficient statistics. Sankhya, Ser. A, 25, 217-244.
6. Berti, P., Consonni, G. and Pratelli, L. (2009). Discussion on the paper: Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci., 23, 179-182.
7. Brown, L.D., Johnstone, I.M. and McGibbon, K.B. (1981). Variance diminishing transformations: A direct approach to total positivity and its statistical applications. J. Amer. Statist. Assoc., 76, 824-832.
8. Casella, G. and George, E. (1992). Explaining the Gibbs sampler. Amer. Statist., 46, 167-174.
9. Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics - Monograph Series, 11. Institute of Mathematical Statistics, Hayward, California.
10. Diaconis, P. and Fill, J. (1990). Strong stationary times via a new form of duality. Ann. Probab., 16, 1483-1522.
11. Diaconis, P. and Fulman, J. (2008). Carries, shuffling and an amazing matrix. Amer. Math. Monthly, 116, 788-803.
12. Diaconis, P. and Fulman, J. (2009). Carries, shuffling, symmetric function. Adv. In Appl. Math., 43, 176-196.
13. Diaconis, P., Khare, K. and Saloff-Coste L. (2008). Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci., 23, 151-178.
14. Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families. Ann. Statist., 7, 269-281.
15. Diaconis, P. and Ylvisaker, D. (1985). Quantifying prior opinion. In Bayesian Statistics, 2 (Valencia, 1983), (J.M. Bernardo, M.H. Degroot, D.V. Lindley, A.F.M. Smith, eds.). North Holland, Amstredam, 133-156.
16. Diaconis, P. and Zabell, S. (1991). Closed form summation for classical distributions: Variations on a theme of deMoivre. Statist. Sci., 61, 284-302.
17. Dudley, R.M. (1989). Real Analysis and Probability. Wadsworth, Belmont, CA.
18. Esch, D. (2003). The skew-t distribution: Properties and computations. Ph.D. Dissertation, Department of Statistics, Harvard University.
19. Fill, J. and Machida, M. (2001). Stochastic monotonicity and realizable monotonicity. Ann. Appl. Probab., 29, 938-978.
20. Givens, C. and Shortt, R. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J., 31, 231-240.
21. Jones, G.L. and Hobert, J.P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16, 312-334.
22. 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.
23. Karlin, S. (1968). Total Positivity. Stanford University Press, Stanford.
24. Khare, K. and Zhou, H. (2009). Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions. Ann. Appl. Probab., 19, 737-777.
25. Lehmann, E. and Romano, J. (2005). Testing Statistical Hypotheses. Springer, New York.
26. Liu, J. (2001). Monte Carlo Strategies in Scientific Computing. Springer-Verlag, New York.
27. Lund, R.B. and Tweedie, R.L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Oper. Res., 20, 182-194.
28. Meyn, S.P. and Tweedie, R.L. (1993). Markov Chains and Stochastic Stability. Spring-er-Verlag, London.
29. Morris, C. (1982). Natural exponential families with quadratic variance functions. Ann. Statist., 10, 65-80.
30. Morris, C. (1983). Natural exponential families with quadratic variance functions: Statistical theory. Ann. Statist. 11, 515-589.
31. Rosenthal, J.S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90, 558-566.
32. Rosenthal, J.S. (1996). Analysis of the Gibbs sampler for a model related to James-Stein estimations. Statist. Comput., 6, 269-275.
33. Rosenthal, J.S. (2002). Quantitative convergence rates of Markov chains: A simple account. Electron. Comm. Probab., 7, 123-128.
34. Saloff-Coste, L. (2004). Total variation lower bounds for finite Markov chains: Wilson's lemma. In Random Walks and Geometry, (V.A. Kaimanovich, K. Schmidt and W. Woess, eds.). Walter de Gruyter GmbH and Co. KG, Berlin, 515-532.
35. Stanley, R. (1989). Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In Graph Theory and Its Applications: East and West (Jinan, 1986). Ann. New York Acad. Sci., 576, New York Acad. Sci., New York, 500-535.
36. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley and Sons, New York.
37. Tierney, L. (1994). Markov chains for exploring posterior distributions (with discus-sion). Ann. Statist., 22, 1701-1762.
38. Wilson, D.B. (2004). Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab., 14, 274-325.