  Quantiles are useful characteristics of random variables that can provide
substantial information on distributions compared with commonly used summary
statistics such as means. In this paper, we propose a Bayesian quantile trend
filtering method to estimate non-stationary trend of quantiles. We introduce
general shrinkage priors to induce locally adaptive Bayesian inference on
trends and mixture representation of the asymmetric Laplace likelihood. To
quickly compute the posterior distribution, we develop calibrated mean-field
variational approximations to guarantee that the frequentist coverage of
credible intervals obtained from the approximated posterior is a specified
nominal level. Simulation and empirical studies show that the proposed
algorithm is computationally much more efficient than the Gibbs sampler and
tends to provide stable inference results, especially for high/low quantiles.
