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Adaptive sampling for Bayesian variable selection


  • David J. Nott
  • Robert Kohn


Our paper proposes adaptive Monte Carlo sampling schemes for Bayesian variable selection in linear regression that improve on standard Markov chain methods. We do so by considering Metropolis--Hastings proposals that make use of accumulated information about the posterior distribution obtained during sampling. Adaptation needs to be done carefully to ensure that sampling is from the correct ergodic distribution. We give conditions for the validity of an adaptive sampling scheme in this problem, and for simulating from a distribution on a finite state space in general, and suggest a class of adaptive proposal densities which uses best linear prediction to approximate the Gibbs sampler. Our sampling scheme is computationally much faster per iteration than the Gibbs sampler, and when this is taken into account the efficiency gains when using our sampling scheme compared to alternative approaches are substantial in terms of precision of estimation of posterior quantities of interest for a given amount of computation time. We compare our method with other sampling schemes for examples involving both real and simulated data. The methodology developed in the paper can be extended to variable selection in more general problems. Copyright 2005, Oxford University Press.

Suggested Citation

  • David J. Nott & Robert Kohn, 2005. "Adaptive sampling for Bayesian variable selection," Biometrika, Biometrika Trust, vol. 92(4), pages 747-763, December.
  • Handle: RePEc:oup:biomet:v:92:y:2005:i:4:p:747-763

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    Cited by:

    1. Eduardo Ley & Mark F.J. Steel, 2009. "On the effect of prior assumptions in Bayesian model averaging with applications to growth regression This article was published online on 30 March 2009. An error was subsequently identified. This not," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(4), pages 651-674.
    2. Ley, Eduardo & Steel, Mark F. J., 2007. "On the effect of prior assumptions in Bayesian model averaging with applications to growth regression," Policy Research Working Paper Series 4238, The World Bank.
    3. Li, Feng & Kang, Yanfei, 2018. "Improving forecasting performance using covariate-dependent copula models," International Journal of Forecasting, Elsevier, vol. 34(3), pages 456-476.
    4. Giordani, Paolo & Kohn, Robert, 2008. "Efficient Bayesian Inference for Multiple Change-Point and Mixture Innovation Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 66-77, January.
    5. Tomi Peltola & Pekka Marttinen & Antti Jula & Veikko Salomaa & Markus Perola & Aki Vehtari, 2012. "Bayesian Variable Selection in Searching for Additive and Dominant Effects in Genome-Wide Data," PLOS ONE, Public Library of Science, vol. 7(1), pages 1-11, January.
    6. Ouysse, Rachida & Kohn, Robert, 2010. "Bayesian variable selection and model averaging in the arbitrage pricing theory model," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3249-3268, December.
    7. Ley, Eduardo & Steel, Mark F.J., 2012. "Mixtures of g-priors for Bayesian model averaging with economic applications," Journal of Econometrics, Elsevier, vol. 171(2), pages 251-266.
    8. Pasanisi, Alberto & Fu, Shuai & Bousquet, Nicolas, 2012. "Estimating discrete Markov models from various incomplete data schemes," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2609-2625.
    9. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    10. Paroli, Roberta & Spezia, Luigi, 2008. "Bayesian inference in non-homogeneous Markov mixtures of periodic autoregressions with state-dependent exogenous variables," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2311-2330, January.
    11. Villani, Mattias & Kohn, Robert & Nott, David J., 2012. "Generalized smooth finite mixtures," Journal of Econometrics, Elsevier, vol. 171(2), pages 121-133.
    12. repec:gam:jecnmx:v:4:y:2016:i:2:p:24:d:68757 is not listed on IDEAS
    13. Tomi Peltola & Pekka Marttinen & Aki Vehtari, 2012. "Finite Adaptation and Multistep Moves in the Metropolis-Hastings Algorithm for Variable Selection in Genome-Wide Association Analysis," PLOS ONE, Public Library of Science, vol. 7(11), pages 1-11, November.
    14. Villani, Mattias & Kohn, Robert & Giordani, Paolo, 2009. "Regression density estimation using smooth adaptive Gaussian mixtures," Journal of Econometrics, Elsevier, vol. 153(2), pages 155-173, December.
    15. Xibin Zhang & Maxwell L. King & Han Lin Shang, 2016. "Bayesian Bandwidth Selection for a Nonparametric Regression Model with Mixed Types of Regressors," Econometrics, MDPI, Open Access Journal, vol. 4(2), pages 1-1, April.
    16. Zhang, Xibin & King, Maxwell L. & Shang, Han Lin, 2014. "A sampling algorithm for bandwidth estimation in a nonparametric regression model with a flexible error density," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 218-234.
    17. Steel, Mark F. J., 2017. "Model Averaging and its Use in Economics," MPRA Paper 81568, University Library of Munich, Germany.
    18. Villani, Mattias & Kohn, Robert & Giordani, Paolo, 2007. "Nonparametric Regression Density Estimation Using Smoothly Varying Normal Mixtures," Working Paper Series 211, Sveriges Riksbank (Central Bank of Sweden).
    19. Fouskakis, D., 2012. "Bayesian variable selection in generalized linear models using a combination of stochastic optimization methods," European Journal of Operational Research, Elsevier, vol. 220(2), pages 414-422.
    20. Li Ma, 2015. "Scalable Bayesian Model Averaging Through Local Information Propagation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 795-809, June.
    21. Nicolas Chopin & Christian Schafer, 2010. "Adaptive Monte Carlo on Multivariate Binary Sampling Spaces," Working Papers 2010-24, Center for Research in Economics and Statistics.
    22. Panagiotelis, Anastasios & Smith, Michael, 2008. "Bayesian identification, selection and estimation of semiparametric functions in high-dimensional additive models," Journal of Econometrics, Elsevier, vol. 143(2), pages 291-316, April.

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