Statistical Science

Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials

Persi Diaconis, Kshitij Khare, and Laurent Saloff-Coste

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We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there.

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Statist. Sci., Volume 23, Number 2 (2008), 196-200.

First available in Project Euclid: 21 August 2008

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Diaconis, Persi; Khare, Kshitij; Saloff-Coste, Laurent. Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials. Statist. Sci. 23 (2008), no. 2, 196--200. doi:10.1214/08-STS252REJ.

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  • [1] Bassetti, F. and Diaconis, P. (2005). Examples comparing importance sampling and the Metropolis algorithm. Illinois J. Math. 50 67–91.
  • [2] Benjamini, I., Berger, N., Hoffman, C. and Mossel, E. (2005). Mixing times of the biased card shuffling and the asymmetric exclusion process. Trans. Amer. Math. Soc. 357 3013–3029.
  • [3] Boyd, S., Diaconis, P., Sun, J. and Xiao, L. (2006). Fastest mixing of Markov chain on a graph and a connection to a maximum variance unfolding problem. SIAM Rev. 48 681–699.
  • [4] Bryc, W., Matysiak, W. and Wesolowski, J. (2008). The bi-Poisson process: A quadratic harness. Ann. Probab. 36 623–646.
  • [5] Consonni, G. and Veronese, P. (1992). Conjugate priors for exponential families having quadratic variance functions. J. Amer. Statist. Assoc. 87 1123–1127.
  • [6] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithm using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157-190.
  • [7] Dyer, M., Goldberg, L., Jerrum, M. and Martin, R. (2005). Markov chain comparison. Probab. Surv. 3 89–111.
  • [8] Griffiths, R. C., Milne, R. K. and Wood, R. (1979). Aspects of correlation in bivariate Poisson distributions and processes. Austral. J. Statist. 21 238–255.
  • [9] Karlis, D. and Meligkotsidou, L. (2005). Multivariate Poisson regression with covariance structure. Stat. Comput. 15 255–265.
  • [10] Khare, K. and Zhou, H. (2008). Rates of convergence of some multivariate Markov chains with poynomial eigenfunctions. Preprint, Dept. Statistics, Stanford Univ.
  • [11] Kontoyiannis, T. and Meyn, S. P. (2003). Spectral theory and limit theory for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 304–362.
  • [12] Kontoyiannis, T. and Meyn, S. P. (2005). Large deviation asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10 61–123.
  • [13] Letac, G. (2000). Stationary sequences with simple joint Poisson distribution. J. Statist. Plann. Inference 90 1–20.
  • [14] Lezaud, P. (2001). Chernoff and Berry–Esseen inequalities for Markov processes. ESAIM Probab. Statist. 5 183–201.
  • [15] Mann, B. (1996). Berry–Esseen central limit theorem for Markov chains. Ph.D. dissertation, Harvard Univ.
  • [16] Rhee, S. Y., Liu, T., Kiuchi, M., Zioni, R., Gifford, R. F., Holmes, S. P. and Shafes, R. W. (2008). Natural variation of HIV group M integrase: Implications for a new class of antiretrovial inhibitors. Preprint, Stanford Univ.