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We present a locally adaptive nonparametric curve fitting method thatoperates within a fully Bayesian framework. This method uses shrinkage priorsto induce sparsity in order-k differences in the latent trend function,providing a combination of local adaptation and global control. Using a scalemixture of normals representation of shrinkage priors, we make explicitconnections between our method and kth order Gaussian Markov random fieldsmoothing. We call the resulting processes shrinkage prior Markov random fields(SPMRFs). We use Hamiltonian Monte Carlo to approximate the posteriordistribution of model parameters because this method provides superiorperformance in the presence of the high dimensionality and strong parametercorrelations exhibited by our models. We compare the performance of three priorformulations using simulated data and find the horseshoe prior provides thebest compromise between bias and precision. We apply SPMRF models to twobenchmark data examples frequently used to test nonparametric methods. We findthat this method is flexible enough to accommodate a variety of data generatingmodels and offers the adaptive properties and computational tractability tomake it a useful addition to the Bayesian nonparametric toolbox.

Locally adaptive smoothing with Markov random fields and shrinkage priors