Nonlocal Priors Research Articles

Computational approach for correcting defocus and suppressing speckle noise in line-field optical coherence tomography images.

The trade-off between transverse resolution and depth-of-focus (DOF) typical for optical coherence tomography (OCT) systems based on conventional optics, prevents "single-shot" acquisition of volumetric OCT images with sustained high transverse resolution over the entire imaging depth. Computational approaches for correcting defocus and higher order aberrations in OCT images developed in the past require highly stable phase data, which poses a significant technological challenge. Here, we present an alternative computational approach to sharpening OCT images and reducing speckle noise, based on intensity OCT data. The novel algorithm uses non-local priors to model correlated speckle noise within a maximum a posteriori framework to generate sharp and noise-free images. The performance of the algorithm was tested on images of plant tissue (cucumber) and in-vivo healthy human cornea, acquired with line-field spectral domain OCT (LF-SD-OCT) systems. The novel algorithm effectively suppressed speckle noise and sharpened or recovered morphological features in the OCT images for depths up to 13×DOF (depth-of-focus) relative to the focal plane.

Bayesian Inference on Hierarchical Nonlocal Priors in Generalized Linear Models

Variable selection methods with nonlocal priors have been widely studied in linear regression models, and their theoretical and empirical performances have been reported. However, the crucial model selection properties for hierarchical nonlocal priors in high-dimensional generalized linear regression have rarely been investigated. In this paper, we consider a hierarchical nonlocal prior for high-dimensional logistic regression models and investigate theoretical properties of the posterior distribution. Specifically, a product moment (pMOM) nonlocal prior is imposed over the regression coefficients with an Inverse-Gamma prior on the tuning parameter. Under standard regularity assumptions, we establish strong model selection consistency in a high-dimensional setting, where the number of covariates is allowed to increase at a sub-exponential rate with the sample size. We implement the Laplace approximation for computing the posterior probabilities, and a modified shotgun stochastic search procedure is suggested for efficiently exploring the model space. We demonstrate the validity of the proposed method through simulation studies and an RNA-sequencing dataset for stratifying disease risk.

Degradation Estimation Recurrent Neural Network With Local and Non-Local Priors for Compressive Spectral Imaging

Degradation Estimation Recurrent Neural Network With Local and Non-Local Priors for Compressive Spectral Imaging

BG2: Bayesian variable selection in generalized linear mixed models with nonlocal priors for non-Gaussian GWAS data

BackgroundGenome-wide association studies (GWASes) aim to identify single nucleotide polymorphisms (SNPs) associated with a given phenotype. A common approach for the analysis of GWAS is single marker analysis (SMA) based on linear mixed models (LMMs). However, LMM-based SMA usually yields a large number of false discoveries and cannot be directly applied to non-Gaussian phenotypes such as count data.ResultsWe present a novel Bayesian method to find SNPs associated with non-Gaussian phenotypes. To that end, we use generalized linear mixed models (GLMMs) and, thus, call our method Bayesian GLMMs for GWAS (BG2). To deal with the high dimensionality of GWAS analysis, we propose novel nonlocal priors specifically tailored for GLMMs. In addition, we develop related fast approximate Bayesian computations. BG2 uses a two-step procedure: first, BG2 screens for candidate SNPs; second, BG2 performs model selection that considers all screened candidate SNPs as possible regressors. A simulation study shows favorable performance of BG2 when compared to GLMM-based SMA. We illustrate the usefulness and flexibility of BG2 with three case studies on cocaine dependence (binary data), alcohol consumption (count data), and number of root-like structures in a model plant (count data).

BGWAS: Bayesian variable selection in linear mixed models with nonlocal priors for genome-wide association studies

BackgroundGenome-wide association studies (GWAS) seek to identify single nucleotide polymorphisms (SNPs) that cause observed phenotypes. However, with highly correlated SNPs, correlated observations, and the number of SNPs being two orders of magnitude larger than the number of observations, GWAS procedures often suffer from high false positive rates.ResultsWe propose BGWAS, a novel Bayesian variable selection method based on nonlocal priors for linear mixed models specifically tailored for genome-wide association studies. Our proposed method BGWAS uses a novel nonlocal prior for linear mixed models (LMMs). BGWAS has two steps: screening and model selection. The screening step scans through all the SNPs fitting one LMM for each SNP and then uses Bayesian false discovery control to select a set of candidate SNPs. After that, a model selection step searches through the space of LMMs that may have any number of SNPs from the candidate set. A simulation study shows that, when compared to popular GWAS procedures, BGWAS greatly reduces false positives while maintaining the same ability to detect true positive SNPs. We show the utility and flexibility of BGWAS with two case studies: a case study on salt stress in plants, and a case study on alcohol use disorder.ConclusionsBGWAS maintains and in some cases increases the recall of true SNPs while drastically lowering the number of false positives compared to popular SMA procedures.

Joint Diagnosis of High-Dimensional Process Mean and Covariance Matrix based on Bayesian Model Selection

Apart from the quick detection of abnormal changes in a process, it is also critical to pinpoint faulty variables after an out-of-control signal. The existing diagnostic procedures mainly focus on the diagnosis of changes in the process mean. This article investigates the joint diagnosis of high-dimensional process mean and covariance matrix based on Bayesian model selection with nonlocal priors. The proposed procedure enjoys two promising features. First, in addition to the isolation of shifted components, it can also provide a probability that the identified components are true, which is very useful for elimination of root causes of abnormal changes. Second, it possesses the model consistency property in the sense that the probability of identifying the true components with shifts approaches one as the sample size increases. The performance comparisons favor the proposed procedure. A real example based on the urban waste water treatment process is provided to illustrate the implementation of the proposed method.

Consistent group selection using nonlocal priors in regression

Consistent group selection using nonlocal priors in regression

Group Sparsity Residual Constraint Image Denoising Model with𝒍1/𝒍2Regularization

Group sparse residual constraint with non-local priors (GSRC) has achieved great success in image restoration producing state-of-the-art performance. In the GSRC model, the [see formula in PDF] norm minimization is employed to reduce the group sparse residual. In recent years, non-convex regularization terms have been widely used in image denoising problems, which have achieved better results in denoising than convex regularization terms. In this paper, we use the ratio of the [see formula in PDF] and [see formula in PDF] norm instead of the [see formula in PDF] norm to propose a new image denoising model, i.e., a group sparse residual constraint model with [see formula in PDF] minimization (GSRC-[see formula in PDF]). Due to the computational difficulties arisen from the non-convexity and non-linearity, we focus on a constrained optimization problem that can be solved by alternative direction method of multipliers (ADMM). Experimental results of image denoising show that the pro-posed model outperforms several state-of-the-art image denoising methods both visually and quantitatively.

Additive Bayesian Variable Selection under Censoring and Misspecification

We discuss the role of misspecification and censoring on Bayesian model selection in the contexts of right-censored survival and concave log-likelihood regression. Misspecification includes wrongly assuming the censoring mechanism to be noninformative. Emphasis is placed on additive accelerated failure time, Cox proportional hazards and probit models. We offer a theoretical treatment that includes local and nonlocal priors, and a general nonlinear effect decomposition to improve power-sparsity trade-offs. We discuss a fundamental question: what solution can one hope to obtain when (inevitably) models are misspecified, and how to interpret it? Asymptotically, covariates that do not have predictive power for neither the outcome nor (for survival data) censoring times, in the sense of reducing a likelihood-associated loss, are discarded. Misspecification and censoring have an asymptotically negligible effect on false positives, but their impact on power is exponential. We show that it can be advantageous to consider simple models that are computationally practical yet attain good power to detect potentially complex effects, including the use of finite-dimensional basis to detect truly nonparametric effects. We also discuss algorithms to capitalize on sufficient statistics and fast likelihood approximations for Gaussian-based survival and binary models.

Low-dose spectral reconstruction with global, local, and nonlocal priors based on subspace decomposition.

Multienergy computed tomography (MECT) is a promising imaging modality for material decomposition, lesion detection, and other clinical applications. However, there is an urgent need to design efficient and accurate algorithms to solve the inverse problems related to spectral reconstruction and improve image quality, especially under low-dose and incomplete datasets. The key issue for MECT reconstruction is how to efficiently describe the interchannel and intrachannel priors in multichannel images. In this model, in order to correlate the similarities of interchannel images and regularize the multichannel images, the global, local, and nonlocal priors are jointly integrated into the low-dose MECT reconstruction model. First, the subspace decomposition method first employs the global low-rankness to map the original MECT images to the low-dimensional eigenimages. Then, nonlocal self-similarity of the eigenimages is cascaded into the optimization model. Additionally, the L0 quasi-norm on gradient images is incorporated into the proposed method to further enhance the local sparsity of intrachannel images. The alternating direction method is applied to solve the optimization model in an iterative scheme. Simulation, preclinical, and real datasets were applied to validate the effectiveness of the proposed method. From the simulation dataset, the new method was found to reduce the root-mean-square error (RMSE) by 42.31% compared with the latest research fourth-order nonlocal tensor decomposition MECT reconstruction (FONT-SIR) method under 160 projection views. The calculation time of an iteration for the proposed method was 23.07% of the FONT-SIR method. The results of material decomposition in real mouse data further confirmed the accuracy of the proposed method for different materials. We developed a method in which the global, local, and nonlocal priors are jointly used to develop the reconstruction model for low-dose MECT, where the global low-rankness and nonlocal prior are cascaded by subspace decomposition and block-matching, and the L0 sparsity is applied to express the local prior. The results of the experiments demonstrate that the proposed method based on subspace improves computational efficiency and has advantages in noise suppression and structure preservation over competing algorithms.

Locally Regularized Collaborative Representation and an Adaptive Low-Rank Constraint for Single Image Superresolution

Learning-based superresolution reconstruction is an efficient image processing technique that has become a popular topic in recent years. Since superresolution is an ill-conditioned problem, appropriate image priors or examples are the key factors for recovering high-quality images with rich details. However, in some current advanced superresolution methods, the mappings established using dictionaries of low- and high-resolution examples cannot effectively reflect the relationship between the low- and high-resolution spaces. Therefore, we introduce a local structure prior to the collaborative representation to constrain the projection matrix; this structure can better represent the nonlinear mapping between the low- and the high-resolution feature spaces. Then, based on the redundancy of similar image patches, a shape-adaptive low-rank constraint is utilized to explore the images’ nonlocal self-similarity, and the local and nonlocal priors complement each other to enhance the recovered image quality. Finally, an iterative optimization algorithm is adopted to solve our proposed superresolution model. Numerous experiments were performed to verify the proposed method, and the results demonstrate its superiority to some state-of-the-art methods both quantitatively and qualitatively.

Concentration of Posterior Model Probabilities and Normalized L0 Criteria

We study frequentist properties of Bayesian and L0 model selection, with a focus on (potentially non-linear) high-dimensional regression. We propose a construction to study how posterior probabilities and normalized L0 criteria concentrate on the (Kullback-Leibler) optimal model and other subsets of the model space. When such concentration occurs, one also bounds the frequentist probabilities of selecting the correct model, type I and type II errors. These results hold generally, and help validate the use of posterior probabilities and L0 criteria to control frequentist error probabilities associated to model selection and hypothesis tests. Regarding regression, we help understand the effect of the sparsity imposed by the prior or the L0 penalty, and of problem characteristics such as the sample size, signal-to-noise, dimension and true sparsity. A particular finding is that one may use less sparse formulations than would be asymptotically optimal, but still attain consistency and often also significantly better finite-sample performance. We also prove new results related to misspecifying the mean or covariance structures, and give tighter rates for certain non-local priors than currently available.

Bayesian penalized model for classification and selection of functional predictors using longitudinal MRI data from ADNI

The main goal of this paper is to employ longitudinal trajectories in a significant number of sub-regional brain volumetric MRI data as statistical predictors for Alzheimer's disease (AD) classification. We use logistic regression in a Bayesian framework that includes many functional predictors. The direct sampling of regression coefficients from the Bayesian logistic model is difficult due to its complicated likelihood function. In high-dimensional scenarios, the selection of predictors is paramount with the introduction of either spike-and-slab priors, non-local priors, or Horseshoe priors. We seek to avoid the complicated Metropolis-Hastings approach and to develop an easily implementable Gibbs sampler. In addition, the Bayesian estimation provides proper estimates of the model parameters, which are also useful for building inference. Another advantage of working with logistic regression is that it calculates the log of odds of relative risk for AD compared to normal control based on the selected longitudinal predictors, rather than simply classifying patients based on cross-sectional estimates. Ultimately, however, we combine approaches and use a probability threshold to classify individual patients. We employ 49 functional predictors consisting of volumetric estimates of brain sub-regions, chosen for their established clinical significance. Moreover, the use of spike-and-slab priors ensures that many redundant predictors are dropped from the model.

Hyperspectral Image Super-Resolution Method Based on Spectral Smoothing Prior and Tensor Tubal Row-Sparse Representation

Due to the limited hardware conditions, hyperspectral image (HSI) has a low spatial resolution, while multispectral image (MSI) can gain higher spatial resolution. Therefore, derived from the idea of fusion, we reconstructed HSI with high spatial resolution and spectral resolution from HSI and MSI and put forward an HSI Super-Resolution model based on Spectral Smoothing prior and Tensor tubal row-sparse representation, termed SSTSR. Foremost, nonlocal priors are applied to refine the super-resolution task into reconstructing each nonlocal clustering tensor. Then per nonlocal cluster tensor is decomposed into two sub tensors under the tensor t-prodcut framework, one sub-tensor is called tersor dictionary and the other is called tensor coefficient. Meanwhile, in the process of dictionary learning and sparse coding, spectral smoothing constraint is imposed on the tensor dictionary, and L1,1,2 norm based tubal row-sparse regularizer is enforced on the tensor coefficient to enhance the structured sparsity. With this model, the spatial similarity and spectral similarity of the nonlocal cluster tensor are fully utilized. Finally, the alternating direction method of multipliers (ADMM) was employed to optimize the solution of our method. Experiments on three simulated datasets and one real dataset show that our approach is superior to many advanced HSI super-resolution methods.

Tensor Completion via Complementary Global, Local, and Nonlocal Priors.

Completing missing entries in multidimensional visual data is a typical ill-posed problem that requires appropriate exploitation of prior information of the underlying data. Commonly used priors can be roughly categorized into three classes: global tensor low-rankness, local properties, and nonlocal self-similarity (NSS); most existing works utilize one or two of them to implement completion. Naturally, there arises an interesting question: can one concurrently make use of multiple priors in a unified way, such that they can collaborate with each other to achieve better performance? This work gives a positive answer by formulating a novel tensor completion framework which can simultaneously take advantage of the global-local-nonlocal priors. In the proposed framework, the tensor train (TT) rank is adopted to characterize the global correlation; meanwhile, two Plug-and-Play (PnP) denoisers, including a convolutional neural network (CNN) denoiser and the color block-matching and 3 D filtering (CBM3D) denoiser, are incorporated to preserve local details and exploit NSS, respectively. Then, we design a proximal alternating minimization algorithm to efficiently solve this model under the PnP framework. Under mild conditions, we establish the convergence guarantee of the proposed algorithm. Extensive experiments show that these priors organically benefit from each other to achieve state-of-the-art performance both quantitatively and qualitatively.

Intrinsic Hyperspectral Image Decomposition With DSM Cues

Intrinsic hyperspectral image decomposition (IHID) aims to recover physical scene properties such as reflectance and illumination from a given hyperspectral image (HSI), which directly respects the physical imaging process and can benefit many HSI processing tasks. It is a severely ill-posed problem and is challenging to solve using HSI alone. Additional geometric information provided by digital surface models (DSMs) can otherwise help immensely. While intrinsic image decomposition for RGB images and RGB-D images has been studied extensively during the past few decades and has seen significant progress, studies of the problem for other types of data, such as HSIs and DSMs, are still needed. It is much more challenging to handle an HSI with hundreds of channels than an RGB image with only three channels. Moreover, compared with RGB-D data, HSIs and DSM data usually have much lower spatial resolutions and more complicated land covers, making it difficult to extend the RGB-D intrinsic image method directly. In this article, we present a novel IHID framework for HSIs with DSM cues. Utilizing spherical-harmonic illumination, we first propose a convenient HSI rendering model with DSM, which describes the interplay of material reflectance, geometric distribution, and environment illumination. Then, we introduce local and nonlocal priors on reflectance that ensure the local smooth and global consistency of recovered reflectance. Experiments on synthetic and real data demonstrate that the proposed method outperforms the state-of-the-art methods and is robust to illumination changes.

On the non-local priors for sparsity selection in high-dimensional Gaussian DAG models

ABSTRACT We consider sparsity selection for the Cholesky factor L of the inverse covariance matrix in high-dimensional Gaussian DAG models. The sparsity is induced over the space of L via non-local priors, namely the product moment (pMOM) prior [Johnson, V., & Rossell, D. (2012). Bayesian model selection in high-dimensional settings. Journal of the American Statistical Association, 107(498), 649–660. https://doi.org/10.1080/01621459.2012.682536] and the hierarchical hyper-pMOM prior [Cao, X., Khare, K., & Ghosh, M. (2020). High-dimensional posterior consistency for hierarchical non-local priors in regression. Bayesian Analysis, 15(1), 241–262. https://doi.org/10.1214/19-BA1154]. We establish model selection consistency for Cholesky factor under more relaxed conditions compared to those in the literature and implement an efficient MCMC algorithm for parallel selecting the sparsity pattern for each column of L. We demonstrate the validity of our theoretical results via numerical simulations, and also use further simulations to demonstrate that our sparsity selection approach is competitive with existing methods.

Approximate Laplace Approximations for Scalable Model Selection

AbstractWe propose the approximate Laplace approximation (ALA) to evaluate integrated likelihoods, a bottleneck in Bayesian model selection. The Laplace approximation (LA) is a popular tool that speeds up such computation and equips strong model selection properties. However, when the sample size is large or one considers many models the cost of the required optimizations becomes impractical. ALA reduces the cost to that of solving a least-squares problem for each model. Further, it enables efficient computation across models such as sharing pre-computed sufficient statistics and certain operations in matrix decompositions. We prove that in generalized (possibly non-linear) models ALA achieves a strong form of model selection consistency for a suitably-defined optimal model, at the same functional rates as exact computation. We consider fixed- and high-dimensional problems, group and hierarchical constraints, and the possibility that all models are misspecified. We also obtain ALA rates for Gaussian regression under non-local priors, an important example where the LA can be costly and does not consistently estimate the integrated likelihood. Our examples include non-linear regression, logistic, Poisson and survival models. We implement the methodology in the R package mombf.

Bayesian group selection with non-local priors

Bayesian group selection with non-local priors

The use of local and nonlocal priors in Bayesian test‐based monitoring for single‐arm phase II clinical trials

Bayesian sequential monitoring is widely used in adaptive phase II studies where the objective is to examine whether an experimental drug is efficacious. Common approaches for Bayesian sequential monitoring are based on posterior or predictive probabilities and Bayesian hypothesis testing procedures using Bayes factors. In the first part of the paper, we briefly show the connections between test-based (TB) and posterior probability-based (PB) sequential monitoring approaches. Next, we extensively investigate the choice of local and nonlocal priors for the TB monitoring procedure. We describe the pros and cons of different priors in terms of operating characteristics. We also develop a user-friendly Shiny application to implement the TB design.