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April 2021 Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities
Jure Vogrinc, Wilfrid S. Kendall
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Ann. Appl. Probab. 31(2): 972-1019 (April 2021). DOI: 10.1214/20-AAP1612

Abstract

For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n1 and as n13 with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions of the order of C1(R) for RWM and C3(R) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H. For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as n1H and as n12+H and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis–Hastings algorithms (possibly depending on a random environment).

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Jure Vogrinc. Wilfrid S. Kendall. "Counterexamples for optimal scaling of Metropolis–Hastings chains with rough target densities." Ann. Appl. Probab. 31 (2) 972 - 1019, April 2021. https://doi.org/10.1214/20-AAP1612

Information

Received: 1 October 2019; Revised: 1 July 2020; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1612

Subjects:
Primary: 60J22
Secondary: 60F05, 65C05

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2021
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