Christensen, Ole F.; Roberts, Gareth O.; Rosenthal, Jeffrey S. Scaling limits for the transient phase of local Metropolis-Hastings algorithms. (English) Zbl 1075.65012 J. R. Stat. Soc., Ser. B, Stat. Methodol. 67, No. 2, 253-268 (2005). Summary: The authors consider high dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well understood and it is now well known how to scale their proposal distribution variances. For the random-walk Metropolis algorithm, convergence during the transient phase is extremely regular – to the extent that the algorithm’s sample path actually resembles a deterministic trajectory. In contrast, the Langevin algorithm with variance scaled to be optimal for stationarity performs rather erratically. We give weak convergence results which explain both of these types of behaviour and practical guidance on implementation based on our theory. Cited in 33 Documents MSC: 65C40 Numerical analysis or methods applied to Markov chains 65C05 Monte Carlo methods 60J22 Computational methods in Markov chains 62D05 Sampling theory, sample surveys Keywords:sampling; numerical examples; Markov chain Monte Carlo methods; Metropolis-Hastings algorithm; transient phase; random-walk Metropolis algorithm; Langevin algorithm; weak convergence PDFBibTeX XMLCite \textit{O. F. Christensen} et al., J. R. Stat. Soc., Ser. B, Stat. Methodol. 67, No. 2, 253--268 (2005; Zbl 1075.65012) Full Text: DOI References: [1] Bedard M., PhD Dissertation (2004) [2] A. Beskos, O. Papaspiliopoulos, G. O. Roberts, and P. N. Fearnhead (2004 ) Exact likelihood-based estimation of diffusion processes . To be published. · Zbl 1100.62079 [3] DOI: 10.1016/S0304-4149(00)00041-7 · Zbl 1047.60065 [4] Christensen O. F., Technical Report 23 (2003) [5] DOI: 10.1111/j.0006-341X.2002.00280.x · Zbl 1209.62156 [6] Ethier S. N., Markov Processes: Characterization and Convergence (1986) · Zbl 0592.60049 [7] Gelman A., Bayesian Statistics 5 pp 599– (1996) · Zbl 0850.62299 [8] Gilks W. R., Markov Chain Monte Carlo in Practice (1996) · Zbl 0832.00018 [9] Grenander U., J. R. Statist. Soc. 56 pp 549– (1994) [10] Hastings W. K., Biometrika 57 pp 97– (1970) [11] DOI: 10.1063/1.1699114 [12] DOI: 10.1111/1467-9469.00115 · Zbl 0931.60038 [13] Neal R., Lect. Notes Statist. 118 (1996) [14] Papaspiliopoulos O., Bayesian Statistics 7 pp 307– (2003) [15] Roberts G. O., Stochast. Stochast. Rep. 62 pp 275– (1998) · Zbl 0904.60021 [16] DOI: 10.1214/aoap/1034625254 · Zbl 0876.60015 [17] DOI: 10.1111/1467-9868.00123 · Zbl 0913.60060 [18] DOI: 10.1214/ss/1015346320 · Zbl 1127.65305 [19] Roberts G. O., Bernoulli 2 pp 341– (1996) [20] G. O. Roberts, and W. K. Yuen (2004 ) Optimal scaling of Metropolis algorithms for discontinuous densities . To be published. [21] DOI: 10.1023/A:1010086427957 · Zbl 0947.60071 [22] DOI: 10.1023/A:1010090512027 · Zbl 0946.60063 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.