zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Rates of convergence of the Hastings and Metropolis algorithms. (English) Zbl 0854.60065
Summary: We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution $\pi$. In the independence case (in $\bbfR^k)$ these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of $\pi$; in the symmetric case (in $\bbfR$ only) we show geometric convergence essentially occurs if and only if $\pi$ has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective.

MSC:
60J05Discrete-time Markov processes on general state spaces
60J20Applications of Markov chains and discrete-time Markov processes on general state spaces
WorldCat.org
Full Text: DOI
References:
[1] AMIT, Y. 1991. On rates of convergence of stochastic relaxation for Gaussian and nonGaussian distributions. J. Multivariate Anal. 38 82 99. · Zbl 0735.60036 · doi:10.1016/0047-259X(91)90033-X
[2] BAXENDALE, P. H. 1993. Uniform estimates for geometric ergodicity of recurrent Markov processes. Technical report, Dept. Mathematics, Univ. Southern California.
[3] BESAG, J. E. and GREEN, P. J. 1993. Spatial statistics and Bayesian computation with. discussion. J. Roy. Statist. Soc. Ser. B 55 25 38. JSTOR: · Zbl 0800.62572 · http://links.jstor.org/sici?sici=0035-9246%281993%2955%3A1%3C25%3ASSABC%3E2.0.CO%3B2-M&origin=euclid
[4] BESAG, J. E., GREEN, P. J., HIGDON, D. and MENGERSEN, K. L. 1995. Bayesian computation Z. and stochastic sy stems with discussion. Statist. Sci. 10 3 66. · doi:10.1214/ss/1177010123
[5] CHAN, K. S. 1993. Asy mptotic behavior of the Gibbs sampler. J. Amer. Statist. Assoc. 88 320 326. JSTOR: · Zbl 0779.62024 · doi:10.2307/2290727 · http://links.jstor.org/sici?sici=0162-1459%28199303%2988%3A421%3C320%3AABOTGS%3E2.0.CO%3B2-T&origin=euclid
[6] HASTINGS, W. K. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97 109. · Zbl 0219.65008 · doi:10.1093/biomet/57.1.97
[7] LUND, R. B. and TWEEDIE, R. L. 1996. Geometric convergence rates for stochastically ordered Markov chains. Math. Oper. Res. To appear. JSTOR: · Zbl 0847.60053 · doi:10.1287/moor.21.1.182 · http://links.jstor.org/sici?sici=0364-765X%28199602%2921%3A1%3C182%3AGCRFSO%3E2.0.CO%3B2-E&origin=euclid
[8] MENGERSEN, K. L. and TWEEDIE, R. L. 1993. Rates of convergence of the Hastings and Metropolis algorithms. Technical Report 93 30, Dept. Statistics, Colorado State Univ.
[9] 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.
[10] MEy N, S. P. and TWEEDIE, R. L. 1993. Markov Chains and Stochastic Stability. Springer, New York.
[11] MEy N, S. P. and TWEEDIE, R. L. 1994. Computable bounds for convergence rates of Markov chains. Ann. Appl. Probab. 4 981 1011. · Zbl 0812.60059 · doi:10.1214/aoap/1177004900
[12] NUMMELIN, E. 1984. General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press. · Zbl 0551.60066
[13] ROBERTS, G. O. and POLSON, N. G. 1994. A note on the geometric convergence of the Gibbs sampler. J. Roy. Statist. Soc. Ser. B 56 377 384. JSTOR: · Zbl 0796.62029 · http://links.jstor.org/sici?sici=0035-9246%281994%2956%3A2%3C377%3AOTGCOT%3E2.0.CO%3B2-V&origin=euclid
[14] ROBERTS, G. O., MENGERSEN, K. L., SCOTT, D. J. and TWEEDIE, R. L. 1995. Bounds on convergence rates in MCMC: a comparison of methods. Unpublished manuscript.
[15] ROBERTS, G. O. and ROSENTHAL, J. S. 1995. Shift-coupling and convergence rates of ergodic averages. Unpublished manuscript.
[16] ROBERTS, G. O. and TWEEDIE, R. L. 1996. Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika. To appear. JSTOR: · Zbl 0888.60064 · doi:10.1093/biomet/83.1.95 · http://www3.oup.co.uk/biomet/hdb/Volume_83/Issue_01/
[17] ROSENTHAL, J. S. 1995. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558 566. JSTOR: · Zbl 0824.60077 · doi:10.2307/2291067 · http://links.jstor.org/sici?sici=0162-1459%28199506%2990%3A430%3C558%3AMCACRF%3E2.0.CO%3B2-2&origin=euclid
[18] SCHERVISH, M. J. and CARLIN, B. P. 1992. On the convergence of successive substitution sampling. Journal of Computational and Graphical Statistics 1 111 127. JSTOR: · doi:10.2307/1390836 · http://links.jstor.org/sici?sici=1061-8600%28199206%291%3A2%3C111%3AOTCOSS%3E2.0.CO%3B2-S&origin=euclid
[19] SCOTT, D. J. and TWEEDIE, R. L. 1996. Explicit rates of convergence of stochastically ordered Markov chains. In Proceedings of Athens Conference on Applied Probability and Time Series. Springer, New York. · Zbl 0858.60060
[20] SMITH, A. F. M. and GELFAND, A. E. 1992. Bayesian statistics without tears: a sampling resampling perspective. Amer. Statist. 46 84 88. JSTOR: · doi:10.2307/2684170 · http://links.jstor.org/sici?sici=0003-1305%28199205%2946%3A2%3C84%3ABSWTAS%3E2.0.CO%3B2-X&origin=euclid
[21] SMITH, A. F. M. and ROBERTS, G. O. 1993. Bayesian computation via the Gibbs sampler Z. and related Markov chain Monte Carlo methods with discussion. J. Roy. Statist. Soc. Ser. B 55 3 24. JSTOR: · Zbl 0779.62030 · http://links.jstor.org/sici?sici=0035-9246%281993%2955%3A1%3C3%3ABCVTGS%3E2.0.CO%3B2-%23&origin=euclid
[22] TIERNEY, L. 1994. Markov chains for exploring posterior distributions with discussion. Ann. Statist. 22 1701 1762. · Zbl 0829.62080 · doi:10.1214/aos/1176325750
[23] TUOMINEN, P. and TWEEDIE, R. L. 1994. Subgeometric rates of convergence of f-ergodic Markov chains. Adv. in Appl. Probab. 26 775 798. JSTOR: · Zbl 0803.60061 · doi:10.2307/1427820 · http://links.jstor.org/sici?sici=0001-8678%28199409%2926%3A3%3C775%3ASROCOM%3E2.0.CO%3B2-S&origin=euclid
[24] BRISBANE, QUEENSLAND 4000 FORT COLLINS, COLORADO 80523 AUSTRALIA