Bayesian trend filtering: adaptive temporal smoothing with shrinkage priors

We present a locally-adaptive nonparametric curve fitting method that we call Bayesian trend filtering. The method operates within a fully Bayesian framework and uses shrinkage priors to induce sparsity in order-k differences in the latent trend function, providing a combination of local adaptation and global control. Using a scale mixture of normals representation of shrinkage priors, we make explicit connections between our Bayesian trend filtering and kth order Gaussian Markov random field smoothing. We use Hamiltonian Monte Carlo to approximate the posterior distribution of model parameters because this method provides superior performance in the presence of the high dimensionality and strong parameter correlations exhibited by our models. We compare the performance of three prior formulations using simulated data and find the horseshoe prior provides the best compromise between bias and precision. We apply Bayesian trend filtering to two benchmark data examples frequently used to test nonparametric methods. We find that this method is flexible enough to accommodate a variety of data generating models and offers the adaptive properties and computational efficiency to make it a useful addition to the Bayesian nonparametric toolbox.