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Imortance Sampling Schemes for Evidence Approximation in Mixture Models

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  • Jeong Eun Lee

    ()
    (Auckland University of Technology)

  • Christian Robert

    ()
    (Université Paris-Dauphine et CREST)

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    Abstract

    The marginal likelihood is a central tool for drawing Bayesian inference about the number of components in mixture models. It is often approximated since the exact form is unavailable. A bias in the approximation may be due to an incomplete exploration by a simulated Markov chain (e.g., a Gibbs sequence) of the collection of posterior modes, a phenomenon also known as lack of label switching, as all possible label permutations must be simulated by a chain in order to converge and hence overcome the bias. In an importance sampling approach, imposing label switching to the importance function results in an exponential increase of the computational cost with the number of components. In this paper, two importance sampling schemes are proposed through choices for the importance function; a MLE proposal and a Rao-Blackwellised importance function. The second scheme is called dual importance sampling. We demonstrate that this dual importance sampling is a valid estimator of the evidence and moreover show that the statistical efficiency of estimates increases. To reduce the induced high demand in computation, the original importance function is approximated but a suitable approximation can produce an estimate with the same precision and with reduced computational workload

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    Bibliographic Info

    Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 2013-42.

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    Length: 22
    Date of creation: Dec 2013
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    Handle: RePEc:crs:wpaper:2013-42

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    Related research

    Keywords: Model evidence; Importance sampling; Mixture models; Marginal likelihood;

    References

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    1. Nicolas Chopin, 2000. "A Sequential Particle Filter Method for Static Models," Working Papers 2000-45, Centre de Recherche en Economie et Statistique.
    2. Robert, Christian P. & Marin, Jean-Michel, 2008. "Approximating the marginal likelihood in mixture models," Economics Papers from University Paris Dauphine 123456789/3692, Paris Dauphine University.
    3. N. Friel & A. N. Pettitt, 2008. "Marginal likelihood estimation via power posteriors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(3), pages 589-607.
    4. Matthew Stephens, 2000. "Dealing with label switching in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 795-809.
    5. Nial Friel & Jason Wyse, 2012. "Estimating the evidence – a review," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 66(3), pages 288-308, 08.
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