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Weak convergence and optimal tuning of the reversible jump algorithm

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  • Gagnon, Philippe
  • Bédard, Mylène
  • Desgagné, Alain

Abstract

The reversible jump algorithm is a useful Markov chain Monte Carlo method introduced by Green (1995) that allows switches between subspaces of differing dimensionality, and therefore, model selection. Although this method is now increasingly used in key areas (e.g. biology and finance), it remains a challenge to implement it. In this paper, we focus on a simple sampling context in order to obtain theoretical results that lead to an optimal tuning procedure for the considered reversible jump algorithm, and consequently, to easy implementation. The key result is the weak convergence of the sequence of stochastic processes engendered by the algorithm. It represents the main contribution of this paper as it is, to our knowledge, the first weak convergence result for the reversible jump algorithm. The sampler updating the parameters according to a random walk, this result allows to retrieve the well-known 0.234 rule for finding the optimal scaling. It also leads to an answer to the question: “with what probability should a parameter update be proposed comparatively to a model switch at each iteration?”

Suggested Citation

  • Gagnon, Philippe & Bédard, Mylène & Desgagné, Alain, 2019. "Weak convergence and optimal tuning of the reversible jump algorithm," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 32-51.
    • Handle: RePEc:eee:matcom:v:161:y:2019:i:c:p:32-51
    DOI: 10.1016/j.matcom.2018.06.007
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    References listed on IDEAS

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    1. S. P. Brooks & P. Giudici & G. O. Roberts, 2003. "Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 3-39, January.
    2. Al-Awadhi, Fahimah & Hurn, Merrilee & Jennison, Christopher, 2004. "Improving the acceptance rate of reversible jump MCMC proposals," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 189-198, August.
    3. Bédard, Mylène & Douc, Randal & Moulines, Eric, 2012. "Scaling analysis of multiple-try MCMC methods," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 758-786.
    4. Sylvia. Richardson & Peter J. Green, 1997. "On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(4), pages 731-792.
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