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Quantiles are useful characteristics of random variables that can providesubstantial information on distributions compared with commonly used summarystatistics such as means. In this paper, we propose a Bayesian quantile trendfiltering method to estimate non-stationary trend of quantiles. We introducegeneral shrinkage priors to induce locally adaptive Bayesian inference ontrends and mixture representation of the asymmetric Laplace likelihood. Toquickly compute the posterior distribution, we develop calibrated mean-fieldvariational approximations to guarantee that the frequentist coverage ofcredible intervals obtained from the approximated posterior is a specifiednominal level. Simulation and empirical studies show that the proposedalgorithm is computationally much more efficient than the Gibbs sampler andtends to provide stable inference results, especially for high/low quantiles.

Fast and Locally Adaptive Bayesian Quantile Smoothing using Calibrated Variational Approximations