Statistical Science

A Geometric Interpretation of the Metropolis-Hastings Algorithm

Louis J. Billera and Persi Diaconis

Full-text: Open access


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.

Article information

Statist. Sci., Volume 16, Number 4 (2001), 335-339.

First available in Project Euclid: 5 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Billera, Louis J.; Diaconis, Persi. A Geometric Interpretation of the Metropolis-Hastings Algorithm. Statist. Sci. 16 (2001), no. 4, 335--339. doi:10.1214/ss/1015346318.

Export citation


  • Diaconis, P. and Hanlon, P. (1992). Eigenanalysis for some examples of the Metropolis algorithm. Contemp. Math. 138 99-117.
  • Diaconis, P. and Ram, A. (2000). Analy sis of sy stematicscan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J. 48 157-190.
  • Diaconis, P. and Saloff-Coste, L. (1998). What do we know about the Metropolis algorithm? J. Comput. Sy stem. Sci. 57 20-36.
  • Dongarra, J. and Sullivan, F., eds. (2000). The top 10 algorithms. Comput. Sci. Engrg. 2.
  • Fishman, G. (1996). Monte Carlo, Concepts, Algorithms and Applications. Springer, New York.
  • Hammersley, J. and Handscomb, D. (1964). Monte Carlo Methods. Chapman and Hall, New York.
  • Hastings, W. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109.
  • Liu, J. (2001). Monte Carlo Techniques in Scientific Computing. Springer, New York.
  • 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.
  • and Teller, E. (1953). Equations of state calculations by fast computing machines. J. Chem. Phy s. 21 1087-1091.
  • Peskun, P. (1973). Optimal Monte Carlo sampling using Markov chains. Biometrika 60 607-612.
  • Roberts, G., Gelman, A. and Gilks, W. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110-120.