## A geometric interpretation of the Metropolis-Hastings algorithm.(English)Zbl 1127.60310

Summary: The Metropolis-Hastings algorithm transforms a given stochastic matrix into a reversible stochastic matrix with a prescribed stationary distribution. We show that this transformation gives the minimum distance solution in an $$L^1$$ metric.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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### References:

 [1] Diaconis, P. and Hanlon, P. (1992). Eigenanalysis for some examples of the Metropolis algorithm. Contemp. Math. 138 99-117. · Zbl 0789.05091 [2] Diaconis, P. and Ram, A. (2000). Analy sis of sy stematicscan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J. 48 157-190. · Zbl 0998.60069 [3] Diaconis, P. and Saloff-Coste, L. (1998). What do we know about the Metropolis algorithm? J. Comput. Sy stem. Sci. 57 20-36. · Zbl 0920.68054 [4] Dongarra, J. and Sullivan, F., eds. (2000). The top 10 algorithms. Comput. Sci. Engrg. 2. [5] Fishman, G. (1996). Monte Carlo, Concepts, Algorithms and Applications. Springer, New York. · Zbl 0859.65001 [6] Hammersley, J. and Handscomb, D. (1964). Monte Carlo Methods. Chapman and Hall, New York. · Zbl 0121.35503 [7] Hastings, W. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109. · Zbl 0219.65008 [8] Liu, J. (2001). Monte Carlo Techniques in Scientific Computing. Springer, New York. · Zbl 0991.65001 [9] Mengersen, K. and Tweedie, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. · Zbl 0854.60065 [10] and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phy s. 21 1087-1091. [11] Peskun, P. (1973). Optimal Monte Carlo sampling using Markov chains. Biometrika 60 607-612. JSTOR: · Zbl 0271.62041 [12] Roberts, G., Gelman, A. and Gilks, W. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110-120. · Zbl 0876.60015
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