Locally Adaptive Smoothing with Markov Random Fields and Shrinkage Priors

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.

Locally adaptive Bayesian birth-death model successfully detects slow and rapid rate shifts

“…We find that both models are capable of inferring variable diversification rates and correctly rejecting variable models in favor of e↵ectively constant models. In general, in line with previous analyses of HSMRF prior distributions [32,29], we see that the HSMRF-based model has higher precision than its GMRF counterpart, with little to no loss of accuracy. In empirical applications, we show that these models are useful for detecting a speciation-rate decline in the Australian gecko clade Pygopodidae and a complex pattern of variation in the rate of infection of HIV subtype A in Russia and Ukraine.…”

Horseshoe‐based Bayesian nonparametric estimation of effective population size trajectories

“…, θ H ) be a vector of parameters that govern the effective population size trajectory N e (t). We propose using a SPMRF model (Faulkner and Minin, 2018) Figure 1: Effective population size trajectory and associated genealogical tree under heterochronous sampling. The top panel shows a continuous effective population size trajectory (gray) and an associated piecewise constant approximation to it.…”

“…A recent method by Faulkner and Minin (2018) uses shrinkage priors in combination with Markov random fields to perform nonparametric smoothing with locally-adaptive properties. This is a fully Bayesian method that does not require the use of knots and avoids the costly computations of inverting dense covariance matrices.…”

“…Computations instead take advantage of the sparsity in the precision matrix of the Markov random field to avoid matrix inversion. Faulkner and Minin (2018) compared different specifications of their shrinkage prior Markov random field (SPMRF) models and found that putting a horseshoe prior on the kth order differences between successive function values had superior performance when applied to underlying functions with sharp breaks or varying levels of smoothness. We refer to the model with the horseshoe prior as a horseshoe Markov random field (HSMRF).…”

Dynamic Shrinkage Processes

“…Model (1) is a Bayesian adaptation of the trend filtering model of Kim et al . () and Tibshirani (), also proposed by Faulkner and Minin (), and includes stochastic volatility (SV) for the observation error variance with a global–local shrinkage prior for the second differences of the conditional mean, Δ 2 β t +1 = ω t . The shrinkage behaviour of ω t determines the path of β t : when ω t is pulled towards zero, β t is locally linear, whereas large innovations correspond to large changes in the slope of β t (Fig.…”