Fast and locally adaptive Bayesian quantile smoothing using calibrated variational approximations

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 study, we propose a Bayesian quantile trend filtering method to estimate the 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.