Optimal Scaling of Mala for Nonlinear Regression

Abstract

We address the problem of simulating efficiently from the posterior distribution over the parameters of a particular class of nonlinear regression models using a Langevin-Metropolis sampler. It is shown that as the number N of parameters increases, the proposal variance must scale as N{-1/3} in order to converge to a diffusion. This generalizes previous results of Roberts and Rosenthal [J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (1998) 255-268] for the i.i.d. case, showing the robustness of their analysis.

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• On the efficiency of pseudo-marginal random walk Metropolis algorithms
• Optimal scalings for local Metropolis--Hastings chains on nonproduct targets in high dimensions
• Diffusion limits of the random walk Metropolis algorithm in high dimensions
• Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions
• Weak convergence of Metropolis algorithms for non-i.i.d. target distributions
• Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior
• Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit