Do Sparsity Promoting Hierarchical Prior Models Plausibly Model Empirical Image Data

Music City Center 

Scale mixtures of normal distributions are a popular family of hierarchical Bayesian models that compromise interpretability, flexibility, and tractability. Recent work has demonstrated that generalized gamma mixtures of normals admit efficient algorithms that allow inference, uncertainty quantification, and hyper-parameter tuning in large, sparse, ill-determined inverse problems. We demonstrate parameter choices that produce popular prior families (Gaussian, Laplace, Student-t), and relate the distinguishability of priors to level sets of low-order moments in the parameter space using the KL and KS distances for power and significance respectively. We test the empirical validity of the hierarchical priors in a series of large imaging data sets. As case studies, we consider benchmark remote sensing, MRI, and image segmentation data sets. We report plausible ranges of parameters under standard representations (Fourier, wavelet, etc.). We find that, relative to the data sets tested, previous computational work has focused on an overly narrow subset of the space of available priors.

Bayesian hierarchical models

Sparse inference

Image data

Scale mixture models

Compressed sensing

Model validation 

Section on Bayesian Statistical Science