Locally adaptive smoothing with Markov random fields and shrinkage priors

TL;DR

This paper introduces a Bayesian nonparametric curve fitting method using shrinkage priors within Markov random fields, enabling local adaptation and global control, with demonstrated superior performance on benchmark data.

Contribution

It develops a novel SPMRF framework linking shrinkage priors to Gaussian Markov random fields, and employs Hamiltonian Monte Carlo for efficient posterior inference.

Findings

Horseshoe prior balances bias and precision effectively.

Method adapts to various data generating models.

Outperforms existing nonparametric methods on benchmarks.

Abstract

We present a locally adaptive nonparametric curve fitting method that operates within a fully Bayesian framework. This method 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 method and kth order Gaussian Markov random field smoothing. We call the resulting processes shrinkage prior Markov random fields (SPMRFs). 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 SPMRF models 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 tractability to make it a useful addition to the Bayesian nonparametric toolbox.