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On the flexibility of the design of multiple try Metropolis schemes

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Abstract

The multiple try Metropolis (MTM) method is a generalization of the classical Metropolis–Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities, show different numerical simulations and discuss the implications of the results.

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Notes

  1. Recall that \(y_i\) are drawn from \(\pi _i(\cdot |x)\) whereas \(x_i^*\) are drawn from \(\pi _i(\cdot |y),\,i=1,\ldots ,k-1,k+1,\ldots N\) and \(x_k^*=x_{t}=x\).

  2. Note that the balance condition is a sufficient but not necessary condition. Namely, the detailed balance ensures invariance. The converse is not true. Markov chains that satisfy the detailed balance condition are called reversible.

  3. However, it is important to remark that high acceptance rates are not a suitable indicator of good performance since, in general, the best acceptance rate is different from 1 (Roberts et al. 1997).

  4. Another simple MTM scheme is the “orientational bias Monte Carlo” (Frenkel and Smit 1996, Chapter 13). In this case, the proposal pdf must be symmetric, i.e., \(\pi (y|x)=\pi (x|y)\), and the weights must be proportional to the target, i.e., \(\omega (y_i)=p(y_i),\,i=1,\ldots ,N\).

  5. Note that, in this work, we have mainly considered scalar variables in order to simplify the treatment and the notation. All the considerations and algorithms contained in this work are also valid for multi-dimensional variables (see, for instance, the last numerical example in Sect. 6.6).

  6. We do not provide the estimated linear correlation because of the moments (as the mean, for instance) of the target do not exist, and it makes difficult a right estimation of the correlation.

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Acknowledgments

We would like to thank the Reviewers for their comments which have helped us to improve the first version of manuscript. Moreover, this work has been partially supported by Ministerio de Ciencia e Innovacin and by the Ministerio de Economía of Spain (project MONIN, ref. TEC-2006-13514-C02- 01/TCM, project COMONSENS, id. CSD2008-00010, project DEIPRO ref. TEC2009-14504-C02-01, project ALCIT, id. TEC2012-38800-C03-01 and project COMPREHENSION, id. TEC2012-38883-C02-01) and Comunidad Autonoma de Madrid (project PROMULTIDIS-CM, ref. S-0505/TIC/0233).

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  1. Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 , Leganés, Madrid, Spain

    Luca Martino & Jesse Read

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  1. Luca Martino
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  2. Jesse Read
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Correspondence to Luca Martino.

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Martino, L., Read, J. On the flexibility of the design of multiple try Metropolis schemes. Comput Stat 28, 2797–2823 (2013). https://doi.org/10.1007/s00180-013-0429-2

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