Sankhya A

, Volume 78, Issue 1, pp 87–104 | Cite as

On Bayesian Quantile Regression Using a Pseudo-joint Asymmetric Laplace Likelihood

  • Karthik SriramEmail author
  • R. V. Ramamoorthi
  • Pulak Ghosh


We consider a pseudo-likelihood for Bayesian estimation of multiple quantiles as a function of covariates. This arises as a simple product of multiple asymmetric Laplace densities (ALD), each corresponding to a particular quantile. The ALD has already been used in the Bayesian estimation of a single quantile. However, the pseudo-joint ALD likelihood is a way to incorporate constraints across quantiles, which cannot be done if each of the quantiles is modeled separately. Interestingly, we find that the normalized version of the likelihood turns out to be misleading. Hence, the pseudo-likelihood emerges as an alternative. In this note, we show that posterior consistency holds for the multiple quantile estimation based on such a likelihood for a nonlinear quantile regression framework and in particular for a linear quantile regression model. We demonstrate the benefits and explore potential challenges with the method through simulations.

Keywords and phrases

Asymmetric Laplace density Bayesian quantile regression Pseudo-likelihood 

AMS (2000) subject classification

Primary 62J02 Secondary 62C10. 


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Copyright information

© Indian Statistical Institute 2015

Authors and Affiliations

  • Karthik Sriram
    • 1
    Email author
  • R. V. Ramamoorthi
    • 2
  • Pulak Ghosh
    • 3
  1. 1.Production and Quantitative Methods AreaIndian Institute of Management AhmedabadAhmedabadIndia
  2. 2.Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  3. 3.Department of Quantitative Methods and Information SystemsIndian Institute of Management BangaloreBangaloreIndia

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