Posterior Consistency of Bayesian Quantile Regression Based on the Misspecified Asymmetric Laplace Density
By Karthikeyan Sriram, Ravi Ramamoorthi, Pulak Ghosh. 2013
131 citations recorded by the source corpus.
We explore an asymptotic justication for the widely used and em- pirically veried approach of assuming an asymmetric Laplace distribution (ALD) for the response in Bayesian Quantile Regression. Based on empirical ndings, Yu and Moyeed (2001) argued that the use of ALD is satisfactory even if it is not the true underlying distribution. We provide a justication to this claim by establishing posterior consistency and deriving the rate of convergence under the ALD misspecication. Related literature on misspecied models focuses mostly on i.i.d. models which in the regression context amounts to considering i.i.d. random covariates with i.i.d. errors. We study the behavior of the posterior for the mis- specied ALD model with independent but non identically distributed response in the presence of non-random covariates. Exploiting the specic form of ALD helps us derive conditions that are more intuitive and easily seen to be satised by a wide range of potential true underlying probability distributions for the response. Through simulations, we demonstrate our result and also nd that the robustness of the posterior that holds for ALD fails for a Gaussian formulation, thus providing further support for the use of ALD models in quantile regression.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9fae2cd8c35ab5637a6565cc5cd24d738c9db12b
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In Artificial Intelligence
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In Statistics and Probability
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Citations in the corpus
- Bayesian Tail Risk Interdependence Using Quantile Regression (2015, methodology): “The Bayesian quantile regression approach (see Yu and Moyeed, 2001; Kottas and Gelfand, 2001; Kottas and Krnjajic, 2009; Sriram et al., 2013) instead considers the distribution of Y | x as belonging to the Asymmetric Laplace distribution family, denoted by ALD (τ, q (x) , σ), with positive σ, whose ”
- Fast Calibrated Additive Quantile Regression (2017, background): “While Sriram, Ramamoorthi, and Ghosh (2013) proved that the resulting posterior concentrates around the true quantile, naively treating the AL density as an adequate probabilistic description of the data is problematic.”
- Calibrating general posterior credible regions (2015, methodology): “A workaround that has been considered by several authors (e.g., Sriram 2015; Sriram et al. 2013; Yu and Moyeed 2001) is to use a (misspecified) asymmetric Laplace likelihood.”
- Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It (2014, background): “settings (Shalizi, 2009), and more specific settings (Sriram et al., 2013); see also Grünwald and Mehta (2016).”