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Published online by Cambridge University Press:  13 July 2016

Andriy Norets*
Brown University
Debdeep Pati
Florida State University
*Address correspondence to Andriy Norets, Associate Professor, Department of Economics, Brown University, Providence, RI 02912; e-mail:


We consider a nonparametric Bayesian model for conditional densities. The model is a finite mixture of normal distributions with covariate dependent multinomial logit mixing probabilities. A prior for the number of mixture components is specified on positive integers. The marginal distribution of covariates is not modeled. We study asymptotic frequentist behavior of the posterior in this model. Specifically, we show that when the true conditional density has a certain smoothness level, then the posterior contraction rate around the truth is equal up to a log factor to the frequentist minimax rate of estimation. An extension to the case when the covariate space is unbounded is also established. As our result holds without a priori knowledge of the smoothness level of the true density, the established posterior contraction rates are adaptive. Moreover, we show that the rate is not affected by inclusion of irrelevant covariates in the model. In Monte Carlo simulations, a version of the model compares favorably to a cross-validated kernel conditional density estimator.

Copyright © Cambridge University Press 2016 

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We thank the editor, the co-editor, and referees for helpful comments. Dr. Pati acknowledges support for this project from the Office of Naval Research (ONR BAA 14-0001) and NSF DMS-1613156.



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