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Optimal scaling of discrete approximations to Langevin diffusions
Author(s) -
Roberts Gareth O.,
Rosenthal Jeffrey S.
Publication year - 1998
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 - 10.1111/1467-9868.00123
Subject(s) - random walk , scaling , mathematics , asymptotically optimal algorithm , limit (mathematics) , scaling limit , dimension (graph theory) , statistical physics , diffusion , function (biology) , langevin dynamics , mathematical optimization , combinatorics , mathematical analysis , physics , statistics , geometry , evolutionary biology , biology , thermodynamics
We consider the optimal scaling problem for proposal distributions in Hastings–Metropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate, independently of the target distribution. The asymptotically optimal acceptance rate is 0.574. We show that, as a function of dimension n , the complexity of the algorithm is O ( n 1/3 ), which compares favourably with the O ( n ) complexity of random walk Metropolis algorithms. We illustrate this comparison with some example simulations.