Weak convergence of Metropolis algorithms for non-i.i.d. target distributions

By Mylene B'edard. 2007

107 citations recorded by the source corpus.

In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234.

Subfields: Condensed Matter Physics, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ef69150b539152bf5fda98c874ad9aed2aaf0ea1

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In Condensed Matter Physics

In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions (2011, background): “1); these papers studied slightly more complicated models, such as products of one-dimensional distributions with different variances and elliptically symmetric distributions [1, 2, 9, 11].”
Optimal scaling of Metropolis algorithms: Heading toward general target distributions (2008, methodology): “For more details, we refer the reader to Bédard (2006a, 2006b, 2006c, 2007).”
Optimal tuning of the hybrid Monte Carlo algorithm (2010, background): “Examples of such problems are contained in the work of Bédard [4, 6, 5], and the papers [9, 27, 35].”
Diffusion limits of the random walk metropolis algorithm in high dimensions (2010, methodology): “0 =MΨ exp(−Ψ(x)) with π0 as constructed above. Frequently in applications, the functional Ψ may not be deﬁned on all of H, but only on a subset Hr ⊂ H for some exponent r >0. For instance, if H=L2([0,1]), the functional Ψ might only act on continuous functions, in which case it is natural to deﬁn”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, background): “For example Metropolis for smooth non iid targets has been addressed, e.g., by Bédard (2007), and in inﬁnite dimensional settings by Beskos et al. (2009).”
Adaptive Gibbs samplers and related MCMC methods (2011, methodology): “For Metropolis algorithms there are various optimal scaling results [37, 41, 9, 10, 4, 42, 43, 46] which provide “prescriptions” of how to do this, though they typically depend on unknown characteristics of π.”
On the stability of sequential Monte Carlo methods in high dimensions (2011, background): “The particular scaling of h was found [5, 43, 44] to provide algorithms that do not degenerate with d .”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “For example Metropolis for smooth non iid targets has been addressed e.g. by B´edard (2007), and in inﬁnite dimensional settings by Beskos et al. (2009).”
Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions (2014, background): “[1] and [5]) so we will leave this for future work for our SMC context.”
Optimal scaling of the random walk Metropolis on unimodal elliptically symmetric targets. (2009, background): “Bedard [2] considers a situation analogous to this, but with R = 1.”