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Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance

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  • Samuel Livingstone

    (Department of Statistical Science, University College London, London WC1E 6BT, UK)

Abstract

We consider a Metropolis–Hastings method with proposal N ( x , h G ( x ) ? 1 ) , where x is the current state, and study its ergodicity properties. We show that suitable choices of G ( x ) can change these ergodicity properties compared to the Random Walk Metropolis case N ( x , h ? ) , either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of G ( x ) can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis.

Suggested Citation

  • Samuel Livingstone, 2021. "Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance," Mathematics, MDPI, vol. 9(4), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:341-:d:495872
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    References listed on IDEAS

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    1. S Livingstone & M F Faulkner & G O Roberts, 2019. "Kinetic energy choice in Hamiltonian/hybrid Monte Carlo," Biometrika, Biometrika Trust, vol. 106(2), pages 303-319.
    2. Craiu, Radu V. & Rosenthal, Jeffrey & Yang, Chao, 2009. "Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1454-1466.
    3. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    4. Søren F. Jarner & Gareth O. Roberts, 2007. "Convergence of Heavy‐tailed Monte Carlo Markov Chain Algorithms," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 781-815, December.
    5. Xifara, T. & Sherlock, C. & Livingstone, S. & Byrne, S. & Girolami, M., 2014. "Langevin diffusions and the Metropolis-adjusted Langevin algorithm," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 14-19.
    6. Mark Girolami & Ben Calderhead, 2011. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(2), pages 123-214, March.
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