Abstract
This chapter is the first of a series on simulation methods based on Markov chains. However, it is a somewhat strange introduction because it contains a description of the most general algorithm of all. The next chapter (Chapter 8) concentrates on the more specific slice sampler, which then introduces the Gibbs sampler (Chapters 9 and 10), which, in turn, is a special case of the Metropolis–Hastings algorithm. (However, the Gibbs sampler is different in both fundamental methodology and historical motivation.)
“What’s changed, except what needed changing?” And there was something in that, Cadfael reflected. What was changed was the replacement of falsity by truth... Ellis Peter, The Confession of Brother Haluin
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Kemeny, J. and Snell, J. (1960). Finite Markov Chains. Van Nostrand, Princeton.
Mykland, P., Tierney, L., and Yu, B. (1995). Regeneration in Markov chain samplers. J. American Statist. Assoc., 90: 233–241.
Schruben, L., Singh, H., and Tierney, L. (1983). Optimal tests for initialization bias in simulation output. Operation. Research, 31: 1176–1178.
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist., 22: 1701–1786.
Tierney, L. (1998). A note on Metropolis-Hastings kernels for general state spaces. Ann. Applied Prob., 8 (1): 1–9.
Tierney, L. and Kadane, J. (1986). Accurate approximations for posterior moments and marginal densities. J. American Statist. Assoc., 81: 82–86.
Tierney, L., Kass, R., and Kadane, J. (1989). Fully exponential Laplace approximations to expectations and variances of non-positive functions. J. American Statist. Assoc., 84: 710–716.
Tierney, L. and Mira, A. (1998). Some adaptive Monte Carlo methods for Bayesian inference. Statistics in Medicine, 18: 2507–2515.
Mira, A. and Geyer, C. (1998). Ordering Monte Carlo Markov chains. Technical report, Univ. of Minnesota.
Mira, A., Moller, J., and Roberts, G. (2001). Perfect slice samplers. J. Royal Statist. Soc. Series B, 63: 583–606.
Tierney, L. and Mira, A. (1998). Some adaptive Monte Carlo methods for Bayesian inference. Statistics in Medicine, 18: 2507–2515.
Brooks, S., Dellaportas, P., and Roberts, G. (1997). A total variation method for diagnosing convergence of MCMC algorithms. J. Comput. Graph. Statist., 6: 251–265.
Grenander, U. and Miller, M. (1994). Representations of knowledge in complex systems (with discussion). J. Royal Statist. Soc. Series B, 56: 549–603.
Phillips, D. and Smith, A. (1996). Bayesian model comparison via jump diffusions. In Gilks, W., Richardson, S., and Spiegelhalter, D., editors, Markov chain Monte Carlo in Practice, pages 215–240. Chapman and Hall, New York.
Brooks, S., Fan, Y., and Rosenthal, J. (2002). Perfect forward simulation via simulation tempering. Technical report, Department of Statistics, Univ. of Cambridge.
Cowles, M. and Rosenthal, J. (1998). A simulation approach to convergence rates for Markov chain Monte Carlo. Statistics and Computing, 8: 115–124.
Corcoran, J. and Tweedie, R. (2002). Perfect sampling from independent Metropolis-Hastings chains. J. Statist. Plana. Inference, 104 (2): 297–314.
Foss, S. and Tweedie, R. (1998). Perfect simulation and backward coupling. Stochastic Models, 14: 187–203.
Mengersen, K. and Tweedie, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist., 24: 101–121.
Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer-Verlag, New York.
Roberts, G. and Tweedie, R. (1995). Exponential convergence for Langevin diffusions and their discrete approximations. Technical report, Statistics Laboratory, Univ. of Cambridge.
Roberts, G. and Tweedie, R. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms Biometrika, 83: 95–110.
Roberts, G. and Tweedie, R. (2004). Understanding MCMC. Springer-Verlag, New York.
Stramer, O. and Tweedie, R. (1999a). Langevin-type models I: diffusions with given stationary distributions, and their discretizations. Methodology and Computing in Applied Probability, 1: 283–306.
Stramer, O. and Tweedie, R. (1999b). Langevin-type models II: Self-targeting candidates for Hastings-Metropolis algorithms. Methodology and Computing in Applied Probability, 1: 307–328.
Stramer, O. and Tweedie, R. (1999b). Langevin-type models II: Self-targeting candidates for Hastings-Metropolis algorithms. Methodology and Computing in Applied Probability, 1: 307–328.
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Robert, C.P., Casella, G. (2004). The Metropolis—Hastings Algorithm. In: Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4145-2_7
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