Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density

Abstract

The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al. (2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for √ n−consistency, (2017) highlighted that the argument was only valid for n α rate for α < 1/2. However, √ n−rate is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we give sufficient conditions for √ n−consistency in the more general setting of Bayesian non-linear quantile regression based on ALD. In particular, we derive √ n−consistency for the Bayesian linear quantile regression. Our approach also enables an interesting extension of the linear case when number of parameters p increases with n, where we obtain posterior consistency at the rate n α for α < 1/2. Sriram and Ramamoorthi

A sandwich likelihood correction for Bayesian quantile regression based on the misspecified asymmetric Laplace density

“…It is more often than not a misspecification of the true underlying likelihood. While posterior consistency still holds under suitable conditions (see Sriram et al 2013), we find that the "coverage property" may not hold. An undesirable consequence would be that a narrow Bayesian credible interval could give a false sense of certainty about a parameter, which is an artifact of a misspecified likelihood rather than an actual gain from the Bayesian approach.…”

“…Proofs of Lemmas 1, 2 and 3 are included in the Appendix and are essentially extensions of ideas from Sriram et al (2013), Kleijn and van der Vaart (2012) and Koenker (2005) respectively. The next lemma establishes the asymptotic connection between the posterior probability and the normal distribution.…”

“…Then, Sriram et al (2013) show under suitable conditions, that the posterior distribution of σ given Y 1 , Y 2 , . .…”

“…We note using Lemma 2(c) of Sriram et al (2013) and Assumption 3b that for sufficiently large n, for some constant C 4 > 0,…”

“…The approach is computationally attractive especially due to the location scale mixture normal representation of ALD (see Kozumi and Kobayashi (2011)), and is known to give posterior consistent estimates even if ALD is a misspecification (see Sriram et al 2013). Therefore, the approach has been useful in many applications (e.g.…”