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Scaling limits for the transient phase of local Metropolis–Hastings algorithms
Author(s) -
Christensen Ole F.,
Roberts Gareth O.,
Rosenthal Jeffrey S.
Publication year - 2005
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/j.1467-9868.2005.00500.x
Subject(s) - metropolis–hastings algorithm , convergence (economics) , algorithm , random walk , transient (computer programming) , scaling , path (computing) , trajectory , langevin dynamics , computer science , statistical physics , mathematics , physics , monte carlo method , statistics , markov chain monte carlo , geometry , operating system , astronomy , economics , programming language , economic growth
Summary. The paper considers 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 algo‐rithm'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.