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Posterior consistency in conditional distribution estimation

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  • Pati, Debdeep
  • Dunson, David B.
  • Tokdar, Surya T.

Abstract

A wide variety of priors have been proposed for nonparametric Bayesian estimation of conditional distributions, and there is a clear need for theorems providing conditions on the prior for large support, as well as posterior consistency. Estimation of an uncountable collection of conditional distributions across different regions of the predictor space is a challenging problem, which differs in some important ways from density and mean regression estimation problems. Defining various topologies on the space of conditional distributions, we provide sufficient conditions for posterior consistency focusing on a broad class of priors formulated as predictor-dependent mixtures of Gaussian kernels. This theory is illustrated by showing that the conditions are satisfied for a class of generalized stick-breaking process mixtures in which the stick-breaking lengths are monotone, differentiable functions of a continuous stochastic process. We also provide a set of sufficient conditions for the case where stick-breaking lengths are predictor independent, such as those arising from a fixed Dirichlet process prior.

Suggested Citation

  • Pati, Debdeep & Dunson, David B. & Tokdar, Surya T., 2013. "Posterior consistency in conditional distribution estimation," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 456-472.
  • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:456-472
    DOI: 10.1016/j.jmva.2013.01.011
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Ryo Kato & Takahiro Hoshino, 2018. "Semiparametric Bayes Instrumental Variable Estimation with Many Weak Instruments," Discussion Paper Series DP2018-14, Research Institute for Economics & Business Administration, Kobe University.
    2. Federico Bassetti & Roberto Casarin & Francesco Ravazzolo, 2015. "Bayesian nonparametric calibration and combination of predictive distributions," Working Paper 2015/03, Norges Bank.
    3. Emine Boz & Gita Gopinath & Mikkel Plagborg-Møller, 2017. "Global Trade and the Dollar," NBER Working Papers 23988, National Bureau of Economic Research, Inc.
    4. repec:taf:jnlasa:v:112:y:2017:i:518:p:806-825 is not listed on IDEAS
    5. Pelenis, Justinas, 2014. "Bayesian regression with heteroscedastic error density and parametric mean function," Journal of Econometrics, Elsevier, vol. 178(P3), pages 624-638.
    6. Laura Liu, 2018. "Density Forecasts in Panel Data Models: A Semiparametric Bayesian Perspective," Papers 1805.04178, arXiv.org.
    7. Debdeep Pati & David Dunson, 2014. "Bayesian nonparametric regression with varying residual density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 1-31, February.
    8. Jing Zhou & Anirban Bhattacharya & Amy H. Herring & David B. Dunson, 2015. "Bayesian Factorizations of Big Sparse Tensors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1562-1576, December.
    9. Seongil Jo & Taeyoung Roh & Taeryon Choi, 2016. "Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 177-206, March.
    10. Norets, Andriy, 2015. "Bayesian regression with nonparametric heteroskedasticity," Journal of Econometrics, Elsevier, vol. 185(2), pages 409-419.
    11. Laura Liu, 2017. "Density Forecasts in Panel Models: A semiparametric Bayesian Perspective," PIER Working Paper Archive 17-006, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 28 Apr 2017.
    12. Antonio Canale & Bruno Scarpa, 2016. "Bayesian nonparametric location–scale–shape mixtures," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 113-130, March.
    13. Ryo Kato & Takahiro Hoshino, 2018. "Semiparametric Bayes Multiple Imputation for Regression Models with Missing Mixed Continuous-Discrete Covariates," Discussion Paper Series DP2018-15, Research Institute for Economics & Business Administration, Kobe University.
    14. Antonio Canale & Bruno Scarpa, 2016. "Bayesian nonparametric location–scale–shape mixtures," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 113-130, March.
    15. Sara Wade & Stephen G. Walker & Sonia Petrone, 2014. "A Predictive Study of Dirichlet Process Mixture Models for Curve Fitting," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 580-605, September.

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