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Optimal scaling of random walk Metropolis algorithms using Bayesian large-sample asymptotics.

Sebastian M Schmon1,2, Philippe Gagnon3

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Summary

This study introduces new tuning guidelines for random walk Metropolis algorithms, improving performance in high dimensions. The research validates these rules under realistic assumptions, ensuring broader applicability in complex simulations.

Keywords:
Bernstein–von Mises theoremLarge-sample theoryMarkov chain Monte CarloOptimal tuningWeak convergence

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Area of Science:

  • Computational Statistics
  • Markov Chain Monte Carlo Methods

Background:

  • High-dimensional limit theorems aid in optimizing random walk Metropolis algorithms.
  • Current methods rely on restrictive product-form target density assumptions, limiting practical use.
  • This necessitates developing tuning rules valid under more realistic conditions.

Purpose of the Study:

  • To investigate optimal scaling in random walk Metropolis algorithms from a large-sample perspective.
  • To derive new tuning guidelines applicable under realistic assumptions, not limited to product-form densities.
  • To address performance deterioration due to correlation structures in high-dimensional targets.

Main Methods:

  • Utilizing a large-sample approach to analyze optimal scaling.
  • Proving weak convergence results under relaxed assumptions.
  • Developing parameter-dimension-dependent tuning guidelines.

Main Results:

  • Weak convergence results are established under realistic assumptions.
  • Novel tuning guidelines are proposed that account for parameter-dimension dependencies.
  • The importance of considering the correlation structure for performance is highlighted.
  • A justification for using an asymptotically exact approximation to the correlation matrix is provided.

Conclusions:

  • The proposed tuning guidelines offer improved performance for random walk Metropolis algorithms in high dimensions.
  • These guidelines are consistent with existing methods when target densities approximate a product form.
  • Accounting for correlation structure is crucial for avoiding performance degradation in non-product-form targets.
  • The study validates practical approximations for initial algorithm runs.