Approximate Bayesian Inference for Quantiles David B. Dunson 1,∗ and Jack A. Taylor 2 1 Biostatistics Branch and 2 Epidemiology Branch, MD A3-03, National Institute of Environmental Health Sciences P.O. Box 12233, Research Triangle Park, NC 27709 ∗ email: dunson1@niehs.nih.gov SUMMARY Suppose data consist of a random sample from a distribution function F Y , which is unknown, and that interest focuses on inferences on θ, a vector of quantiles of F Y . When the likelihood function is not fully specified, a posterior density cannot be calculated and Bayesian infer- ence is difficult. This article considers an approach which relies on a substitution likelihood characterized by a vector of quantiles. Properties of the substitution likelihood are investi- gated, strategies for prior elicitation are presented, and a general framework is proposed for quantile regression modeling. Posterior computation proceeds via a Metropolis algorithm that utilizes a normal approximation to the posterior. Results from a simulation study are presented, and the methods are illustrated through application to data from a genotoxicity experiment. KEY WORDS: Comet assay; Nonparametric; Median regression; Order constraints; Prior elicitation; Quantile regression; Single cell electrophoresis; Substitution likelihood. 1