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Optimal scaling of the random walk Metropolis algorithm under \(L^p\) mean differentiability. (English) Zbl 1401.65007

Summary: In this paper, we consider the optimal scaling of high-dimensional random walk Metropolis algorithms for densities differentiable in the \(L^p\) mean but which may be irregular at some points (such as the Laplace density, for example) and/or supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension \(d\) goes to \(\infty\). As the log-density might be nondifferentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of the last author et al. [Ann. Appl. Probab. 7, No. 1, 110–120 (1997; Zbl 0876.60015)]. This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

MSC:

65C05 Monte Carlo methods
60F05 Central limit and other weak theorems

Citations:

Zbl 0876.60015
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References:

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