Bayesian Quantile Trend Filtering on Graphs using Shrinkage Priors

Quantiles are useful characteristics of random variables that can provide substantial information of 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 on graphs. We introduce general shrinkage priors for graph differences to induce locally adaptive Bayesian inference on trends. Introducing so-called shadow priors with multivariate truncated distribution for local scale parameters and mixture representation of the asymmetric Laplace distribution, we provide a simple Gibbs sampling algorithm to generate posterior samples. We also develop variational Bayes approximation to quickly compute point estimates (e.g. posterior means). The numerical performance of the proposed method is demonstrated through simulation study with time series data, application of quantile regression and robust spatial quantile smoothing.