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Optimal Scaling of Discrete Approximations to Langevin Diffusions

Gareth O. Roberts and Jeffrey S. Rosenthal
Journal of the Royal Statistical Society. Series B (Statistical Methodology)
Vol. 60, No. 1 (1998), pp. 255-268
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2985986
Page Count: 14
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Optimal Scaling of Discrete Approximations to Langevin Diffusions
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Abstract

We consider the optimal scaling problem for proposal distributions in Hastings--Metropolis algorithms derived from Langevin diffusions. We propose 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(n1/3), which compares favourably with the O(n) complexity of random walk Metropolis algorithms. We illustrate this comparison with some example simulations.

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