Horseshoe priors for edge-preserving linear Bayesian inversion

In many large-scale inverse problems, such as computed tomography and image deblurring, characterization of sharp edges in the solution is desired. Within the Bayesian approach to inverse problems, edge-preservation is often achieved using Markov random field priors based on heavy-tailed distributions. Another strategy, popular in statistics, is the application of hierarchical shrinkage priors. An advantage of this formulation lies in expressing the prior as a conditionally Gaussian distribution depending of global and local hyperparameters which are endowed with heavy-tailed hyperpriors. In this work, we revisit the shrinkage horseshoe prior and introduce its formulation for edge-preserving settings. We discuss a sampling framework based on the Gibbs sampler to solve the resulting hierarchical formulation of the Bayesian inverse problem. In particular, one of the conditional distributions is high-dimensional Gaussian, and the rest are derived in closed form by using a scale mixture representation of the heavy-tailed hyperpriors. Applications from imaging science show that our computational procedure is able to compute sharp edge-preserving posterior point estimates with reduced uncertainty

A Horseshoe Pit mixture model for Bayesian screening with an application to light sheet fluorescence microscopy in brain imaging

Finding parsimonious models through variable selection is a fundamental problem in many areas of statistical inference. Here, we focus on Bayesian regression models, where variable selection can be implemented through a regularizing prior imposed on the distribution of the regression coefficients. In the Bayesian literature, there are two main types of priors used to accomplish this goal: the spike-and-slab and the continuous scale mixtures of Gaussians. The former is a discrete mixture of two distributions characterized by low and high variance. In the latter, a continuous prior is elicited on the scale of a zero-mean Gaussian distribution. In contrast to these existing methods, we propose a new class of priors based on discrete mixture of continuous scale mixtures providing a more general framework for Bayesian variable selection. To this end, we substitute the observation-specific local shrinkage parameters (typical of continuous mixtures) with mixture component shrinkage parameters. Our approach drastically reduces the number of parameters needed and allows sharing information across the coefficients, improving the shrinkage effect. By using half-Cauchy distributions, this approach leads to a cluster-shrinkage version of the Horseshoe prior. We present the properties of our model and showcase its estimation and prediction performance in a simulation study. We then recast the model in a multiple hypothesis testing framework and apply it to a neurological dataset obtained using a novel whole-brain imaging technique

Group Inverse-Gamma Gamma Shrinkage for Sparse Regression with Block-Correlated Predictors

Heavy-tailed continuous shrinkage priors, such as the horseshoe prior, are widely used for sparse estimation problems. However, there is limited work extending these priors to predictors with grouping structures. Of particular interest in this article, is regression coefficient estimation where pockets of high collinearity in the covariate space are contained within known covariate groupings. To assuage variance inflation due to multicollinearity we propose the group inverse-gamma gamma (GIGG) prior, a heavy-tailed prior that can trade-off between local and group shrinkage in a data adaptive fashion. A special case of the GIGG prior is the group horseshoe prior, whose shrinkage profile is correlated within-group such that the regression coefficients marginally have exact horseshoe regularization. We show posterior consistency for regression coefficients in linear regression models and posterior concentration results for mean parameters in sparse normal means models. The full conditional distributions corresponding to GIGG regression can be derived in closed form, leading to straightforward posterior computation. We show that GIGG regression results in low mean-squared error across a wide range of correlation structures and within-group signal densities via simulation. We apply GIGG regression to data from the National Health and Nutrition Examination Survey for associating environmental exposures with liver functionality.Comment: 44 pages, 4 figure

Dynamic Shrinkage Priors for Large Time-varying Parameter Regressions using Scalable Markov Chain Monte Carlo Methods

Time-varying parameter (TVP) regression models can involve a huge number of coefficients. Careful prior elicitation is required to yield sensible posterior and predictive inferences. In addition, the computational demands of Markov Chain Monte Carlo (MCMC) methods mean their use is limited to the case where the number of predictors is not too large. In light of these two concerns, this paper proposes a new dynamic shrinkage prior which reflects the empirical regularity that TVPs are typically sparse (i.e. time variation may occur only episodically and only for some of the coefficients). A scalable MCMC algorithm is developed which is capable of handling very high dimensional TVP regressions or TVP Vector Autoregressions. In an exercise using artificial data we demonstrate the accuracy and computational efficiency of our methods. In an application involving the term structure of interest rates in the eurozone, we find our dynamic shrinkage prior to effectively pick out small amounts of parameter change and our methods to forecast well.Comment: Keywords: Time-varying parameter regression, dynamic shrinkage prior, global-local shrinkage prior, Bayesian variable selection, scalable Markov Chain Monte Carlo JEL Codes: C11, C30, E3, D3

Dimension-free Mixing for High-dimensional Bayesian Variable Selection

Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called "two-stage drift condition" to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis