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Density Estimation for the Metropolis–Hastings Algorithm

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  • M. Sköld
  • G. O. Roberts

Abstract

. Kernel density estimation is an important tool in visualizing posterior densities from Markov chain Monte Carlo output. It is well known that when smooth transition densities exist, the asymptotic properties of the estimator agree with those for independent data. In this paper, we show that because of the rejection step of the Metropolis–Hastings algorithm, this is no longer true and the asymptotic variance will depend on the probability of accepting a proposed move. We find an expression for this variance and apply the result to algorithms for automatic bandwidth selection.

Suggested Citation

  • M. Sköld & G. O. Roberts, 2003. "Density Estimation for the Metropolis–Hastings Algorithm," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 699-718, December.
    • Handle: RePEc:bla:scjsta:v:30:y:2003:i:4:p:699-718
    DOI: 10.1111/1467-9469.00359
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    1. Adjemian, Stéphane & Juillard, Michel & Karamé, Fréderic & Mutschler, Willi & Pfeifer, Johannes & Ratto, Marco & Rion, Normann & Villemot, Sébastien, 2024. "Dynare: Reference Manual, Version 6," Dynare Working Papers 80, CEPREMAP.
    2. Tao Hu & Xinyan Zhu & Lian Duan & Wei Guo, 2018. "Urban crime prediction based on spatio-temporal Bayesian model," PLOS ONE, Public Library of Science, vol. 13(10), pages 1-18, October.
    3. Adjemian, Stéphane & Bastani, Houtan & Juillard, Michel & Karamé, Fréderic & Mihoubi, Ferhat & Mutschler, Willi & Pfeifer, Johannes & Ratto, Marco & Rion, Normann & Villemot, Sébastien, 2022. "Dynare: Reference Manual Version 5," Dynare Working Papers 72, CEPREMAP, revised Mar 2023.
      • Stéphane Adjemian & Houtan Bastani & Michel Juillard & Frédéric Karamé & Ferhat Mihoubi & Willi Mutschler & Johannes Pfeifer & Marco Ratto & Sébastien Villemot & Normann Rion, 2023. "Dynare: Reference Manual Version 5," PSE Working Papers hal-04219920, HAL.
      • Stéphane Adjemian & Houtan Bastani & Michel Juillard & Frédéric Karamé & Ferhat Mihoubi & Willi Mutschler & Johannes Pfeifer & Marco Ratto & Sébastien Villemot & Normann Rion, 2023. "Dynare: Reference Manual Version 5," Working Papers hal-04219920, HAL.
    4. Mathieu Langlard & Fabrice Lamadie & Sophie Charton & Johan Debayle, 2021. "Bayesian Inference of a Parametric Random Spheroid from its Orthogonal Projections," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 549-567, June.
    5. Bee, Marco & Espa, Giuseppe & Giuliani, Diego, 2015. "Approximate maximum likelihood estimation of the autologistic model," Computational Statistics & Data Analysis, Elsevier, vol. 84(C), pages 14-26.
    6. Fabian Krüger & Sebastian Lerch & Thordis Thorarinsdottir & Tilmann Gneiting, 2021. "Predictive Inference Based on Markov Chain Monte Carlo Output," International Statistical Review, International Statistical Institute, vol. 89(2), pages 274-301, August.
    7. Alexey Miroshnikov & Evgeny Savelev, 2019. "Asymptotic properties of parallel Bayesian kernel density estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 771-810, August.

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