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Accept–reject Metropolis–Hastings sampling and marginal likelihood estimation

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  • Siddhartha Chib
  • Ivan Jeliazkov

Abstract

We describe a method for estimating the marginal likelihood, based on Chib (1995) and Chib and Jeliazkov (2001), when simulation from the posterior distribution of the model parameters is by the accept–reject Metropolis–Hastings (ARMH) algorithm. The method is developed for one‐block and multiple‐block ARMH algorithms and does not require the (typically) unknown normalizing constant of the proposal density. The problem of calculating the numerical standard error of the estimates is also considered and a procedure based on batch means is developed. Two examples, dealing with a multinomial logit model and a Gaussian regression model with non‐conjugate priors, are provided to illustrate the efficiency and applicability of the method.

Suggested Citation

  • Siddhartha Chib & Ivan Jeliazkov, 2005. "Accept–reject Metropolis–Hastings sampling and marginal likelihood estimation," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 59(1), pages 30-44, February.
    • Handle: RePEc:bla:stanee:v:59:y:2005:i:1:p:30-44
    DOI: 10.1111/j.1467-9574.2005.00277.x
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    1. Train,Kenneth E., 2009. "Discrete Choice Methods with Simulation," Cambridge Books, Cambridge University Press, number 9780521766555, September.
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    3. Rufo, M.J. & Martín, J. & Pérez, C.J., 2010. "New approaches to compute Bayes factor in finite mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3324-3335, December.
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    5. Arnab Kumar Maity & Sanjib Basu & Santu Ghosh, 2021. "Bayesian criterion‐based variable selection," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(4), pages 835-857, August.
    6. Choudhary, Vidyanand & Currim, Imran & Dewan, Sanjeev & Jeliazkov, Ivan & Mintz, Ofer & Turner, John, 2017. "Evaluation Set Size and Purchase: Evidence from a Product Search Engine," Journal of Interactive Marketing, Elsevier, vol. 37(C), pages 16-31.
    7. Joshua C.C. Chan & Rodney Strachan, 2014. "The Zero Lower Bound: Implications for Modelling the Interest Rate," Working Paper series 42_14, Rimini Centre for Economic Analysis.
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    9. Asai, Manabu, 2009. "Bayesian analysis of stochastic volatility models with mixture-of-normal distributions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(8), pages 2579-2596.
    10. Frederico M. Almeida & Vinícius D. Mayrink & Enrico A. Colosimo, 2023. "Bayesian solution to the monotone likelihood in the standard mixture cure model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 77(3), pages 365-390, August.
    11. Nakajima, Jouchi & Omori, Yasuhiro, 2009. "Leverage, heavy-tails and correlated jumps in stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2335-2353, April.
    12. Ehlers, Ricardo S., 2012. "Computational tools for comparing asymmetric GARCH models via Bayes factors," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(5), pages 858-867.
    13. Michael L. Polemis & Mike G. Tsionas, 2023. "The environmental consequences of blockchain technology: A Bayesian quantile cointegration analysis for Bitcoin," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 28(2), pages 1602-1621, April.
    14. Hedibert Freitas Lopes, 2014. "A Tutorial on the Computation of Bayes Factors," Business and Economics Working Papers 200, Unidade de Negocios e Economia, Insper.
    15. Jouchi Nakajima & Yasuhiro Omori, 2007. "Leverage, Heavy-Tails and Correlated Jumps in Stochastic Volatility Models (Revised in January 2008; Published in "Computational Statistics and Data Analysis", 53-6, 2335-2353. April 2009. )," CARF F-Series CARF-F-107, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    16. Michael J. Daniels & Arkendu S. Chatterjee & Chenguang Wang, 2012. "Bayesian Model Selection for Incomplete Data Using the Posterior Predictive Distribution," Biometrics, The International Biometric Society, vol. 68(4), pages 1055-1063, December.
    17. Chan, Joshua & Strachan, Rodney, 2012. "Estimation in Non-Linear Non-Gaussian State Space Models with Precision-Based Methods," MPRA Paper 39360, University Library of Munich, Germany.
    18. Jouchi Nakajima, 2008. "EGARCH and Stochastic Volatility: Modeling Jumps and Heavy-tails for Stock Returns," IMES Discussion Paper Series 08-E-23, Institute for Monetary and Economic Studies, Bank of Japan.

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